It's a variety of things, but yes to increase size and strength you do need to eventually lift heavier things, with the best method of doing so considered to be progressive overload. There are much better resources than me but to give some basics: while bodybuilding is not a direct translation to strength (see strongman competitions and their body types), it absolutely does correlate and bodybuilders are actually quite strong, their goals are just usually way different than strongmen. Anecdotally, I have done both, and prefer strength training mentally, but my body prefers bodybuilding as I tended to get a lot of minor injuries with strength training because it is a little more risky (assuming proper form across both disciplines).

Basically, you need to continue "overloading" what your body can lift over time, and you do this by increasing reps, sets, and weights. You don't need a ton of weight to do this for your upper body, but legs and back you'll eventually want some plates and whatnot. You can also overload in different ways, higher reps and more sets will help increase both your endurance and strength, though moreso the former. Over time, you will be able to add weight, add reps, add sets, and you will absolutely gain in size and strength.

The other key items are hitting the weights frequently and diet. If you want to get bigger you need to eat a surplus of calories and gain weight, "bulking." You can do this either cleanly (eating healthy) or do a dirty bulk (obviously, not clean eating), and you'll gain either way, though typically you'll add more fat in a dirty bulk. Then, you can then lose some of the weight later while "cutting," the most miserable time for a bodybuilder, where you try to maintain your prior workout intensity, weight, etc, but now operating at a calorie deficit in order to keep the muscle while losing the fat.

Strength programs will typically call for fewer sets/reps and to increase the weight as fast as you safely can while bodybuilding is best pursued with higher reps, though I believe it usually caps at sets of 12 or so reps before you start adding weight for optimal size increases. In my experience, this makes bodybuilding more time consuming as well, but you can recover faster.