As an EU citizen, you don't have any time limit on remaining in the UK necessarily, under EU freedom of movement rules. Now freedom of movement technically applies only to workers, but practically it means that you can enter the "EU" line in the passport queue, and nobody will bother asking you any questions. It also means that you could remain once your study had finished, simply by looking for a job ( you could register at the jobcentre to prove this if really needed)

The specific rules for students depends a little on how long you will be staaying for.

Staying abroad for up to 3 months

As an EU national, you have the right to study in another EU country. If you stay there for less than 3 months, all you will need is a valid identity card or passport.

Staying abroad for more than 3 months

You have the right to live in any EU country for the duration of your studies if you:

• are enrolled in an approved educational establishment
• have sufficient income (from any source) to live without needing income support
• have comprehensive health insurance cover there.

National authorities may not require your income to be above the level that would qualify you for basic income support.

You could lose your right to stay in the country if you finish your studies and cannot prove you are in work or still have sufficient resources to support yourself.

Relaxed provides a link to this overview of what freedom of movement means. For you, it means you can just saunter in pretty casual and go to your course.

As @CMaster explained, you have a right to stay for as long as you like in the UK and you're not required to prove you will leave as there are many ways for you to make sure you are entitled to a longer stay (like finding a job…). Unlike non-EU nationals, you are also entitled to change your mind about this at any time, there is no requirement for you to obtain a specific visa in advance depending on the purpose of your stay. If you stay longer than three months, you could in principle be asked then, but only then, to show that you fulfil some conditions (i.e. that you are either working, studying, have sufficient resources or live with an EU citizen who does).

In principle, you therefore should not be asked any question. It's not only that border guards don't bother but they are not supposed to assess anything apart from the genuineness of your ID and citizenship and possibly that you do not present a threat to national security. Your plans and their credibility are immaterial and don't need to be discussed like they are for third-country nationals (including US citizens).

In practice, I frequently go to the UK on my ID card from another EU country and I never had to answer any question whatsoever. You line up in front of the EU/EEA citizen booth, along with British people, some border guard looks at your ID, scans it, and looks at your face, maybe grumbles a few words and you are on your way (that's for immigration, customs is later, incidentally). Nobody will want to see your ticket or ask where you are going or what you want to do. A passport isn't needed either, a Dutch ID card would be enough (but a passport is completely fine too of course).

The only bureaucratic difficulty I encountered when I moved from Germany to the UK to work there was that after a few weeks in my new job I got a form in which I was supposed to fill in the dates of every single previous stay in the UK. I had been there before, but that had been many years ago and obviously I didn't remember precise dates. I would expect that giving approximations to such a stupid and pointless question does not have any consequences, though.

The only other issue you might have is with health insurance. You will probably be covered (for free!) by NHS even when moving from the US to the UK to study there, but I am not sure and you should check this. In any case I would expect that once your studies are over and you only stay in the UK for a few weeks for fun you will technically not be covered by NHS. And NHS is certainly not responsible for you while traveling elsewhere in Europe. So if you get diagnosed with something serious during that time, or have a bad accident, you might be in serious trouble if you don't get some other form of health insurance.

One issue you might run into after your stay: In some cases a health insurer may require proof that you were covered by health insurance previously. If you lived in the UK there isn't really anything to prove because (almost, see above) everyone is automatically covered by NHS. As a consequence, if a health insurer insists on proof, as happened in the case of my daughter, it's not at all clear what to do. If you think this might happen to you, be sure to get some form of proof before you leave the country. When I tried this from Germany, I failed. In the UK the process usually involves calling special phone numbers that are not easily accessible from outside the country, and of course it's also harder to get information from abroad.