The designers expected people to play the game just as they had in 2e, with only the barest minimum of strategy change. This is why Clerics got the "spontaneous conversion" stuff, and decent attack actions--Clerics were expected to be Brother Bactine almost all the time, so they were given a bunch of benefits to cushion that blow. Instead, players looked at all those benefits and the inefficiency of in-combat healing and said, "Why would I ever heal when I could do a bunch of other, more-useful things instead?" This type of disconnect, between "we expect players to keep playing just like in 2e, so we should adjust things assuming that they will" and "analyze the game for what it actually rewards and then pursue that," characterizes about half of the problems with 3rd edition's balance.
About a third of its issues center on the failure to test scaling beyond relatively low levels. People have mentioned E6 and how 6th level is about the last level where things still work right--well, that's because that's about as far as they playtested, from what I've been told. They presumed that patterns which held fine up to that point would hold fine forever after, and...they don't. You can see this most strongly with the Fighter class: having tons of feats was supposed to be extremely powerful, but in practice because of the long and tiny-fiddly feat chains, most Fighters actually require a TON of optimization just to be halfway decent, let alone great. A failure to test and examine how feats actually worked when spooled out over many levels meant the designers had a false idea of how valuable a feat-slot was.
Most of the remainder is down to simply not considering whether features played nicely together, best exemplified by the Monk. The Monk is supposed to be a mobile combatant flying around the battlefield, doing cool martial arts things. The problem is, in order to do the best damage you can in 3rd ed, you must stand completely still. They then stapled on several other random grab-bag features that don't actually cohere together into any meaningful whole; the 3rd edition Monk just frankly sucks and there's very little you can do to fix it that doesn't effectively become "replace it with a different class." (As an example, there are feats that allow characters to count their Monk levels as advancing psionic powers, essentially suturing together Monk and whatever psionic class you prefer.)
The final little bit--which has a disproportionate impact for its size--is the presence of specific spells and feats that blow the game balance wide open, semi-related to my second paragraph above. Infamous examples being the spell glitterdust and the feat Natural Spell. The former is simultaneously excellent crowd control and completely defeats enemy stealth or invisibility when it's successful (and it's easy to make it successful). The latter allows essentially every Druid ever to become an incredible powerhouse, taking on animal forms that rival the Fighter's power while also being able to cast spells. That feat is almost single-handedly responsible for the D in "CoDzilla," and it seems pretty clear that the designers just....never considered how powerful they were making the Druid class.
So...yeah. The designers of 3e expected an unchanged culture of play from 2e, they didn't test the game beyond low levels, and they more than once failed to actually make cohesive design goals for some of the classes. Druid was incoherent but incredibly powerful (suturing together "pet class," "shapeshifting class," "full nature spellcaster," and "summoning specialist" all in one package!) while Monk was incoherent in a way that made it weak.