There is a difference between an implicit claim and an assumption. An assumption is an unstated point that must be true for an argument or chain of reasoning to work. One makes an assumption. An implicit claim is something that is claimed within a claim. The person doesn't assume the implicit claim is true, he implicitly claims it is true.
For example: "Despite his low popularity, Newt Gingritch may yet take the Republican nomination." Here there is an implicit claim that Newt Gingritch has low popularity. We are in fact claiming it, but we're doing it in an indirect way that tries to disguise the fact that we're claiming it.
The way language works, stating a proposition is vouching for it. If I said, "Your wife is cheating on you", no rational person would respond, "Oh, you've just stated a logical proposition that may or may not be true". It's understood that the person is claiming that the proposition is true by stating it in that way. It's an explicit claim.
A claim can be implied by stating the proposition in a way that doesn't vouch for its truth. One common way to do this is to make a claim that cannot be true (or wouldn't be sensible to claim) unless some other proposition is true. That proposition is implicitly claimed.
In the specific case of advertising, the term "implicit claim" is used to mean a claim that a reasonable person might conclude from a slogan or claim. For example, if I sell my product with a jingle that includes the line, "kills the germs that cause bad breath", a reasonable person might infer that my product reduces bad breath. In fact this isn't claimed at all and this use is different from the way the term is used in philosophical debate and other fields. This is really something that isn't claimed at all.