Usually you'll see someone spamming grab if they're desperate and just fishing for damage, especially if they have a tether. Plus I tend to see a lot of Ness users getting grab happy, though considering just how powerful his grabs are it's understandable. Can be a bit of a nightmare if you're using a heavyweight.
It's not too hard to counter act overly grabby players. If you've got some decent close range options you can get by easy, like with Dedede using Inhale or his crouch attack will usually make a good punish. If you don't have anything of that sort, you're best off just backing off and thinking over the approach.
Plus don't be disheartened if you end up losing a match from the grabs. In the rematch you'll at least have the knowledge of what kind of approach they'll go for and you can adjust accordingly. If all else fails, just pick a character with good ranged normals.
(Also let me know if anything I'm saying is bad advice, I'm far from an expert.)