I am going to apply for one semester of study abroad at a more prominent university than the one I am currently enrolled in. One of my true motivations to apply is to secure letters of recommendations in case I'll apply for a graduate school later on. Should I mention this motivation? My personal statement will be read by both sides involved. Will the mention make it sound that the letters from my university are worth less? Does the reason in itself count as one of legitimate motives to go?
No, you shouldn't mention securing letters of recommendation in your personal statement. It sounds a little too mercenary, like you are more interested in bureaucratic outcomes like letters than you are in actually learning. It may hurt the feelings of faculty from your home institution who read it, and it's not really what the other institution is looking for in a personal statement.
On the other hand, you're right that this experience could really help in securing compelling letters, and I think it's sensible for you to take this into account in your (private) decision making. If you currently attend a strong research university, then getting letters from an even stronger research university probably won't help much. However, there are many students at teaching-focused schools that don't send many students to graduate school in any given field, and very rarely to the top departments. That makes it harder for faculty to write strong letters of recommendation, because the comparison group is not so good: even if somebody writes "this is the best student I've seen in 20 years", it's not clear that this is in itself a strong enough statement. (It's still possible to write a compelling letter if the recommender can offer enough concrete, impressive details, but it makes writing the letter harder and requires the recommender to know the applicant really well.) If a faculty member can easily compare you with lots of students attending a range of good to great graduate schools, that makes it easier for them to write a useful letter.
So if you're studying in a department that rarely or never sends students to the graduate programs you'd like to attend, then it's very much worth trying to get a letter from someone who has a broader perspective on the applicant pool. Otherwise it's not so important. But either way, you shouldn't mention it in your personal statement.