It depends which games you're talking about, and who's playing them. Allow me to elaborate;
Some games stand the test of time on two simple merits before nostalgia is taken into account. Those merits would be genuinely good design, and by extension, revolutionising the design of other games in their genre.
The two franchises most often credited with those merits for their earlier games are Mario & Sonic, and rightly so. Boot up Mario 3 or Sonic 2 and you'll find they stand up incredibly well next to their modern incarnations (and sometimes surpass them - I'm looking at you, hedgehog).
This isn't just because they're so well designed. It's because those designs deeply affected designs of platformers of their time, and continue to do so to this day. When we play modern platform/action adventure games, they've been influenced so greatly by Mario/Sonic that going back to those older titles doesn't feel like such a huge leap backwards in time.
Also helping these games is their visual aesthetic. They're games from the 16-bit era, and yet they're still pleasing to look at. It's because they were designed with the limitations and strengths of their systems in mind and, through use of bright colours, imaginative scenery and sharp, well defined sprites, still look great, even without nostalgiavision.
The earlier Halo games hold up for the same reasons. Whether you love them or hate them, the changes they made are still used in the industry today. Dedicated buttons for grenades and melee attacks, regenerating shields/health and the two weapon limit (among other things) are pretty much standard for modern FPS. So, when you boot up a copy of Halo 2, the only thing holding it back from being a "modern" FPS is its last-gen graphics. Whether you think that's cause for praise for Halo or condemnation for modern FPS is up to you, but it's still true.
Some games don't hold up so well because the genre moves on - other games in their genre innovate and revolutionise, and so when you boot up Resident Evil on the PS1 or Mortal Kombat on the SNES, they feel like archaic remnants of a time gone by, because they lack modern features that bettered their genres.
Let me elaborate using Resident Evil as an example. At the time, the game really was revolutionary - its influence can be felt in virtually every horror game of its era. Such things as pushing a statue from a second floor balcony in order to collect an item from its shattered remains on the floor below had never been done before. Hell, it had never been possible before. But now, it's commonplace and outdated. Resident Evil's maligned "tank" control system was a necessity because it used pre-rendered images to create high quality back grounds. With static images forming the background in a 3D game, the camera angle needed to change from one fixed position to another every time the character moved off-screen. To feasibly control a character in this situation, the controls needed "Forward", "Back", "Left" and "Right" to be constant directions, unaffected by the camera's position, lest the character end up running in circles as the camera angle changed.
These days, such trickery isn't necessary. Modern consoles and PCs can render near-photorealistic environments without breaking a sweat. Controllers have dual analogue sticks as standard, negating the need for "stand and turn" controls. Developers figured out real time weapon change & item usage, on-screen mini-maps and so on, negating the need to pop open your inventory every few seconds. As storage space on discs became bigger and compression techniques more effective, games could pack in more and more locations, eliminating the need for games to enforce backtracking as a way to pad their length. Resident Evil had none of these more modern innovations and thus, doesn't hold up, except perhaps in the eyes of players already adjusted to and accepting of its flaws.
This doesn't mean games that haven't aged well can't have redeeming qualities that make them still worth a look in a time that left them behind, though. To use the cliché example, Shadows of the Colossus looks like shit, has horrible controls, a worse camera, far too few changes in scenery and is incredibly dull at some points. But, for all the reasons we're familiar with already, it's still worth experiencing.
You can apply this logic to just about any title from the past to judge how well it holds up. Just be careful who you point out a game's flaws to - as accurate as you may be, pointing out that Silent Hill 2 looks, plays and sounds like a sedated bovine is likely to earn you a few stern words.