Well... no one's responding so I'll give it a whirl.
First though, I gotta' say the wording to your question, "what issue did it solve that was a problem in 3.x?" sounds vaguely like a trap. I blame the internet, though.
For me, 3.x is a lot of fun if it's house-ruled to the point where it is very dissimilar to 3.x. I don't know if this is a problem or not. A lot of the house rules I instituted were fun, but fairly broken in favour of the PCs and myself having a good time (most of these rules were in regards to combat). I continually tried to unbreak these house rules and slowly got closer and closer to something that was fun and not... so unstable. I was going to run another game of D&D (which would be another experiment in house rules) when I was linked by a friend to 4e info. It really seemed like a lot of what I was trying to accomplish, but hardcoded into the rules combatwise.
After playing the demo, I really felt that I wouldn't have to tweak and bend into really weird ways these rules to make a game with consistently fun combat.
Skill challenges solve a problem I was blind to. When I first read about them, I thought, "Those sound dumb. I'll probably never use them. They sound like a tool for teaching new players how to role-play. Cool idea, but probably not for me." ...I was partially correct. The main problem skill challenges 'solve' is giving players a tool to roleplay in a plot effective way using a combination of their own ingenuity, character skill, and luck. The system may not be the most perfect, but it (was for me) a really fun step in the right direction. I had always thought that traps and puzzles worked in such a way that the rogue and spell caster w/ useful knowledge were the only (I'm being super general here, I know) characters who could contribute to certain puzzles and traps. Skill challenges 'solve' this problem by making it a fallacy. Every party member can be viable in role-play scenarios and contribute to the team's efforts. That sounds good to me.
Also, the demo changed my view of defenders. They're hardcore. Not that they weren't before, and not that any other position is lame, but I think defenders are better equipped featurewise to be nifty.
I know that that is all opinion and very much focal to my experiences. I hope it somehow helped you with what you wanted to know.