Full disclosure up front: I'm a guitarist who plays some solo/accompanying gigs, so a proper bass player I ain't. That said, I've doubled on bass on a gig or two, and I've spent a fair amount of time on the theory. The following is basically how I learned to do what I'm able to do.
If you want to get into jazz, I'd say first thing to do is get a good fake book. I like The Real Book a lot myself, and around these parts it's the go-to book for gigs. Everybody shows up carrying one of these. There is a bass clef edition of the book which might come in handy for you, but you're not really in this for the melody. What you want is a book of different chord progressions to practice improvising on, and the treble clef edition will serve that purpose just as well. Just steer clear of Eb or Bb editions, as those are for horn players and they'll land you in strange keys.
I'd also grab some backing tracks (this question asks about jazz piano but my answer applies to you as well). They're a great practice tool.
Once you've got your book and tracks, I'd spend some time tinkering with tunes. Find one that looks a little simpler ("Autumn Leaves" and "Blue Bossa" are great starter tunes, for example) and play! Easy, right?
Okay, I'll elaborate. There are a few main categories of things you'll need to learn.
There's a lot to learn about rhythm, but if you're trying to learn jazz I'd start with swing type tunes. In those cases, usually, the bass player will be be playing "in two" or "in four". "In two" means playing the first and third beats of the measure. "In four" means playing on each quarter note. Try to wrap your ears around that listening a bit, and try playing the rhythm a bit too, without worrying too much about notes. I like to practice with a metronome set to half my desired tempo, and play as if it's hitting the 2 and 4 (like most jazz drummers will do). Think "One, beep, three, beep, one, beep, three, beep."
The bass player tends to drive the rhythm of a jazz group more even than the drummer, so try to get your ears around that rhythmic feel as best you can.
If you want to grow into fusion- or funk-influenced playing, there are a lot of resources out there that can help you learn about syncopation. I honestly did it more by feel, listening to players I liked and copping their licks. Listen to lots and lots of recordings!
Bass players don't need to know about chords, right? Wrong!
I found that learning to play bass lines was the greatest single thing I could have done for my music theoretical knowledge. Not only do you have to be able to follow "the changes" (that is to say, the chord changes. Following, of course, the appropriate harmonic rhythm), you need to know about chord tones and chord-scale relationships in order to "walk" convincingly.
Start by playing only the root notes of the chords as written. C for Cm, D for Dm7b5, G for G7#5 and so on. If you don't know your way around the fretboard well enough to pick those notes out, that's going to be a bit of a project, but it's a necessary step. There are a great many different resources out there for that, from charts to wacky mnemonic systems to video lessons. Find what works best for you and learn it!
You'll want to learn how to construct chords. Bass players often spend a lot of time bouncing between the 1 and the 5 of a chord (that is, the root of the chord and the note a fifth up from that root). On a normal major or minor chord, that's no problem. But the 5 of a half-diminished chord is flatted! And the 5 of an altered dominant chord might be flatted or sharped. What does that mean?
Chord-scale relationships are also really important. What does a Dm7 imply? Usually the D Dorian mode, and it usually leads to a G7 (usually G Mixolydian), which usually leads to a C major (usually C Ionian). All those are actually modes of the C major scale. These things are important to know, and while they seem incredibly daunting at first, learning them will make quick work of the meatier tunes down the road.
A further note about scales: The bulk of jazz harmony is based on four scales. They are, in roughly descending order of importance, the major scale, the melodic minor scale, the diminished scale, and the whole tone scale. The modes of these scales are also used, so it's important to know those. "D dorian" means the second mode of the C major scale. "G altered" means the seventh mode of the Ab melodic minor scale. And so on and so forth... There are also pentatonic scales, which have only five notes and are generally formed of a subset of one of the above scales or modes. Bebop scales are derived from the above modes but with an extra note stuffed in between a couple of the others as a passing tone. There's also the blues scale, with a similar addition to the minor pentatonic scale. I personally prefer to identify a chord with one of the first four scales, then to make note choices with the latter scales in mind. Your mileage may vary.
Substitutions are also useful to know. Look up and learn what a tritone substitution is. The Tadd Dameron turnaround is a related concept, and very useful. If you're feeling ambitious, try to figure out how Bird changes are related to a twelve-bar blues, or start digging into Coltrane changes.
With regards to harmony, Jamey Aebersold's reference sheets "Jazz Nomenclature" and "The Scale Syllabus" will be good resources for you. I printed copies and kept them handy for years. I'd also highly recommend Mark Levine's "The Jazz Theory Book".
Once you've learned a little about the theory behind what notes you can play, you'll have to choose the notes you want to play. Smooth transitions are nice. In the first couple bars of a blues progression, for example, moving from a Bb7 chord to an Eb7 chord, you might transition to the Eb7 from the third of the Bb7 chord, a D. Slide in from a half-step below, like. Or you could jump up to the 5th of the Bb7 chord, an F, and drop in stepwise from above. Or you may want to remain static while the rest of the band moves around you, playing a pedal point. Or you could employ sequencing to give a structured feel to notes that don't actually belong to the chord or key you're playing in at all. The sky's the limit.
Much of this will come from listening, and much of this will come from practice. You're going to make mistakes, and you're going to discover beautiful things.
There's a lot to digest here, I know, but always remember, it's very possible to learn this stuff. People have done it, and will continue to do it for years to come. Listen to and play along with as many recordings as you can. Good traditional bass players to listen to include Charles Mingus, Paul Chambers, or Ray Brown. On electric bass, you can hardly go wrong with the likes of Jaco Pastorius, Stanley Clarke, or Steve Swallow.
Start with simple tunes like "Blue Bossa", "Autumn Leaves", blues tunes like "Straight, No Chaser", "Blue Monk", "Mr. P.C.". Work your way up to some ballads (they're slower, but tend to have more difficult chord changes) like "Body and Soul", "Misty", or "God Bless The Child". If you want to play groovier stuff, think about tunes like "Cantaloupe Island", "Mercy, Mercy, Mercy", and "The Chicken".
And always always follow your ears. If you hear somebody whose sound you like, there's a good chance they played on a whole bunch of albums. Ron Carter, for example, has appeared on over 2,500 albums. Listen, try to play along when you can, try to understand what they're doing.