I feel like I understand the concept of advantage (rolling attack to hit) for surrounding an opponent. However, In one game I am playing, the DM imposes disadvantage (rolling attack to hit) for the person being surrounded. In other words, me being surrounded doesn't just give the people attacking me an advantage in their attack rolls. When it comes time for me to attack, now I am rolling at disadvantage.

I am okay with this in concept. It makes sense. Question: Is this part of what other people are doing too? Is there anywhere you would send me to read about it in the D&D rules?

Being surrounded doesn't grant advantage or disadvantage in the rules. Disadvantage is most likely a DM's houserule. I never seen it in use as a player or DM.

For advantage is an optional rule for Flanking and also some monsters have a trait called Pack Tactics that can do so.

I’ve not seen it used, either. But it seems reasonable, so long as you knew in advance, and it applies to the enemies, as well. 

I think it's a little bit too strong. Look, you're flanked! Oh, that's right, we have advantage and you? Disadvantage.

The optional rule for Flanking is in the DMG: https://www.dndbeyond.com/sources/dmg/running-the-game#OptionalRuleFlanking

It is not in the PHB, the SRD or the Basic Rules.

This makes flanking extremely powerful. I'm not sure your DM appreciates all the consequences of this. Attacking via saving throw becomes much better (impacting martials much more than spellcasters), positioning abilities and spells become more valuable, being outnumbered is even more of a serious threat, huddling together becomes a decent strategy, which in turn makes AOEs more powerful, etc, etc.

In short, the game's balance is based on certain assumptions. This is messing with one of those assumptions to a degree that significantly impacts overall balance. My first impression is that he wants to make weaker enemies like goblins and kobolds more threatening when they swarm you. If that's the case, I'd houserule the creatures themselves rather than a blanket combat rule.

And if the primary argument is that "it makes sense," I'd point out that D&D is as much a game as it is a fantasy simulation. What makes more sense is to follow the rules of a game because the game was designed around those rules.

The closest thing in RAW would be ranged attacks in close combat -- you have disadvantage on ranged attacks if an enemy is in melee with you. They're possibly either expanding on or misinterpreting that rule. The section I linked to is in the Basic Rules section on combat, and that chapter has all the essentials on how combat works.

I was running a game where we tried this flanking rule, it was horrendous, the PCs were hitting far more than they should and if I had used it with the creatures, it would have been a TPK encounter. So we ditched it after that one go.

It isn't fair to the DM or PC, the PC can counteract the DM's amount of creatures by hiring underlings. That way it would either cancel out or give the PCs advantage all the time if they can hire enough underlings! 

So long as your DM understands that those same rules apply when YOU flank enemies, then at least it's fair. I do agree with the general consensus of others, though, that this won't make for a very satisfying combat experience.