I primarily play piano, but I keep an electric guitar around to play with from time to time. I don't think this is right - the bridge seems to be tilted instead of flush with the guitar. This is a Squier stratocaster, if that helps.
Your suspicion is correct -- that's not set up properly. If you take off the rectangular back panel of your guitar, you should see a number of springs that connect the metal block underneath your tremolo bridge to a "claw" piece that is screwed into the body of the guitar.
These springs should fully counteract the tension created by the strings on the other side of the instrument. You can adjust this tension by screwing the claw further into the body, and by adding more springs.
If you've got three springs in the back, that should be enough to balance strings of a typical gauge, but heavier strings may require four springs.
So presuming you have enough springs and everything in the back is in good working order, all you need to do is adjust the claw to your preferences. This kind of bridge requires enough spring tension such that it is pulled back to rest against the body of the guitar, but can be more tight to emulate a hardtail bridge (with no movement of the tremolo) or less tight if you do plan to install the tremolo arm and have some play when you move it.
The exact setting would be up to you, but if this is just a beater guitar that you want to play around with from time to time, you will probably want to add enough spring tension such that the bridge is not going to be going anywhere, even with strings installed--like a hardtail. Much less maintenance than managing a guitar with a floating or close-to-floating trem.
Also, in the present configuration, you are probably having action and intonation issues from the string saddles being closer and higher up than they should be positioned. So, you definitely want to get this fixed up.
Excellent answer above! I would also like to add, in my experience with these strat type vibrato systems, they're actually Vibrato systems and not Tremolos. But, my experience is this, after 20 some years and hundreds and hundreds of stratocaster type guitars through my workshop.. not all of the Claw Springs are the same.
Depending on the maker of the guitar, or more importantly the country of manufacture, not all springs are made of the same materials. I've had some Squier strats with very soft and flexible springs on the spring claws. They almost have a sort of blueish look to the metal of the springs. Even with the spring claw screwed all the way close to the body of the guitar, the bridge would still lift up one me.
When I'd go to bend a string, the springs would still let the bridge lift up off the body. It was set with just a set of 9 gauge strings, low action and three springs on the claw. So if I had say an open bass note ringing and bending a single string like say a g string, the force from the bend would pull the bridge up with the bend, and throw my bass note flat. So not all springs are made the same.
Some have softer material they're made with where others have a stronger metal material. Also, if your springs are nickel plated or chrome plated, that helps add strength to the springs so they do not become to flexible. The unfinished springs, the blue looking kind, I think just don't hold the tension at all that well on some guitars. I usually would replace the cheaper springs out with some good nickel plated ones and the problem was solved.
Another thing I want to add is this, if you don't ever want to mess with the springs again, or worry about cheap ones letting your bridge plate ever lift from bending strings.. you can do one simple thing. Simply block the bridge in place. You take a block of wood that fit between the body cavity of where the vibrato block is and the body of your guitar, and place a wooden block between the two, that fits snug. This way, if you ever do some hard bending, the wooden block will not allow the metal block of the vibrato plate to be lifted up from the springs. Its a cool way to stop any tuning issues with those strat vibrato systems.
And if you want to ever use your vibrato you can just install your whammy bar and lift up and let the block slip out. And you're good to go again. After years of having so many guitars with different strat floating vibratos, I've found not all the springs are the same. Some are thin gauge of wire and others are bigger, some are soft metals and some are hard metals. Anyways, just a tip for anyone having the same problems.