Native speakers have been arguing about this for centuries. There are two schools of thought.
The conjunction theory
The conjunction theory, most famously advocated by Robert Lowth (1710–1787), claims that than is a conjunction that introduces a new clause with its own subject. So, you say:
He can walk faster than I can walk.
For brevity, you can omit the repeated can walk:
He can walk faster than I.
Since I is the subject of the omitted verb, it should be in the subjective case. Therefore the objective case, as in "He can walk faster than me," is nonsense, quod erat demonstrandum.
The preposition theory
The preposition theory says that than can serve as a conjunction or as a preposition, whichever you need. When used as a preposition, its object naturally takes the objective case:
He can walk faster than me.
As far as I know, no blood has ever been spilled over this dispute, but it has been surprisingly acrimonious.
Advocates of the conjunction theory claim that logic is on their side, as well as prior usage such as the phrase holier than thou, which appears in the King James Bible. One argument in favor of the conjunction theory is that the choice of subjective or objective case makes a useful distinction if than is always regarded as a conjunction. For example, “Priya likes chapatis more than I” is short for “Priya likes chapatis more than I like chapatis” while “Priya likes chapatis more than me” is short for “Priya likes chapatis more than she likes me.”
Advocates of the preposition theory claim that logic is on their side, as well as convenience and prior usage in such authoritative writers as Shakespeare and Samuel Johnson (see the Wikipedia article or a book on grammar myths).
Advocates of the preposition theory like to point out that the same people who advocate the conjunction theory also advocate other grammar myths such as the “rule” against ending a sentence with a preposition and the “rule” against splitting an infinitive. Perhaps this is an ad hominem fallacy, and perhaps it’s an insight into something very misguided deep in the minds of the conjunction-theory advocates.
You must take sides
Since you are learning English as a foreign language, you probably would prefer to stay out of a debate about such a small point of grammar, affecting only about six words in the entire language. Sorry, that is not an option. You must choose a side. Once you say faster than I or faster than me, everyone will know which side you're on. The room will turn silent. People will look at each other with knowing glances. Friends who favor the other side will probably shun you for life when they find out. Friends who favor the side you've chosen will feel a new and enriched bond of loyalty to you.*
Let me close, then, by mentioning that I favor the preposition theory, and so should you.
* Don’t think you can weasel out of this by always spelling out the implied clause explicitly. That will avoid making a choice, but you’ll soon find it verbose and tiresome. And no one likes a waffler.