Learning scales and chord inversions
I disagree with him that you shouldn't "just memorize" them. Learning scales and chord inversions is all about repetition, repetition, repetition. I don't think I know any teachers who would say "no, don't memorize scales by themselves". Scales are the basis of EVERYTHING you play on the piano (or most other pitched instruments, too, for that matter). If you want to be able to play like he can and improvise your way through anything, you must learn the scales and commit them to memory.
Learning chord progressions and improvisation
Here's where he is correct. If you want to learn how to play any song and improvise anything in any key, the best way is to expose yourself to music and understand what's happening. This is how I, too, taught myself how to play almost anything in any key in just a few years.
In order to that, however, you first need to learn a few basics.
1) Learn how to build chords. If someone says, "play a C chord", you should be able to immediately hit a C chord. If someone says, "play a Gm-add6", you should be able to immediately hit a G minor (add 6) chord. There are countless resources on the internet that have charts and diagrams you can learn chords from.
2) Learn your scale degrees. If you're playing in the key of C and someone says "play a IV chord", you should know to play an F chord. If someone says "play a vi chord", you should know to play an A minor. Once you understand the concept of what a "IV (4) chord" or a "V (5) chord" is, you will be able to grasp the idea of chord progressions much more easily.
3) Learn your basic chord progressions. Learn the "rooms" of the keys. When he says to familiarize yourself with the "rooms", he's talking about learning all the standard chords associated to a given key. That key has a "room", with all the chords in it that you can choose. For example, in the key C, your room typically will consist of things like a I chord, a IV chord, V chord, a vi chord, a II chord, and so on. If someone says to play a song in the key of C, you should be familiar with these "standard" chords you will play with.
Here's an example. Using ONLY the chords mentioned above, here are some songs you can play in the key of C (the "room" of C):
"Let it Go" - I V vi IV*
"Oh, When the Saints" - I V I IV I V I
"My Bonnie Lies Over the Ocean" - I IV I V I
*this chord progression applies to a LOT of pop songs-- "Let it Go", "Don't Stop Believing", "Country Roads, Take Me Home", "I'm Yours", etc.
4) Start immersing yourself in music (preferably music with fairly basic chord progressions, to start out). Find some guitar chords for a particular song. Since you learned how scale degrees work in Step #2, you should be able to look at that chord sheet and tell what the chord progression is.
For example, if you know the song is in the key of G and the chords are G C G D G, you know that the chord progression is I IV I V I. Practice the song until you are very, very comfortable with playing the chord progression.
5) Here's where the magic begins. Remember that I IV I V I we just figured out in Step #4? We can take that progression and apply it to any key. Maybe we want to play the song in the key of D. Well, if we play a I IV I V I pattern in the key of D, we end up with D G D A D.
Over time, you still hopefully start to develop an ear for chord progressions and be able to tell which scale degree is being played at any given time. As your teacher alluded to, each note and chord has a specific purpose in the scale/key, and it's up to you to know how to use each one (which is largely what Steps 1-5 are to help you with). Once you can do that, you can start playing along with songs without any written music. That is the point at which you can start improvising.
Disclaimer: While the process is fairly straightforward, my explanation you see here is quite oversimplified. This isn't a "learn it all in a week" thing. It's a "Take a step here, practice. Take another step, practice." thing. Don't be surprised or frustrated if it takes you a while to get to Step 5. It's a process of learning and growing.
Disclaimer #2: I write this answer under the assumption you already are a competent piano player. If you are still in the process of learning which keys play which notes, please learn that first. You will only frustrate yourself by trying to jump ahead. Again, it's a process of learning and growing.
Disclaimer #3: If want to learn how to improvise because you can't read music, I strongly caution you. Yes, many phenomenal pianists can't actually read music. But, I would still highly encourage you to learn to read music. It will help you in the long run.
EDIT: I forgot to make a final comment on chord inversions. Once you have locked down your ability to play with songs using the chord progressions, you can start experimenting with inversions and finding out what sounds the best. Some songs lend themselves to using a 2nd inversion. Some songs lend themselves to using a 1st inversion. Some songs lend themselves to switching it up. Assuming you have an ear for music, play around with it and find something that suits your fancy. Develop your personal style.
