What makes the E-Trade talking baby commercials effective?
These ads are to some extent like the GEICO "caveman" ads. They are meant to some extent to underline the idea that using E*Trade is something that is very easy to do. By having babies in the ads, they get across the idea that even a baby could use this service.
However, not all of the ads actually explicitly put this message across. Instead, they are effective because they are amusing. They do show the babies doing things like investing, but then they also put them in other adult situations like having a baby have a girlfriend who accuses him of cheating on her. The idea is simply that people will respond to the innate absurdity of the ads.
So, the ads are effective because they tell us (sometimes explicitly and sometimes not) that using this service is very easy. But they are also effective simply because they are amusing and will make people remember the E*Trade brand.