So here’s how I’d go about trying to remember けど. Bunpro’s lesson starts by pointing out that けど is a conjunction particle. As a particle, it doesn’t have a meaning by itself, as normal nouns and adjectives do, but gives structure to a sentence by showing how various elements relate to each other. As a conjunction, it joins two separate statements together. So in your example
その赤いTシャツが好き。
I like that red T-shirt.
and
青いTシャツを買う。
I will buy the blue T-shirt.
are two separate statements joined by けど to get the full sentence
その赤いTシャツが好きだけど、青いTシャツを買う。
I like that red T-shirt, but I will buy the blue T-shirt.
Because 好き is a na-adjective, which are often treated similarly to nouns, だ must be placed between it and けど here.
けど has the nuance of implying a contrast between the statements that it links - I like the red T-shirt, so you might think I would purchase it, but in fact I will buy the blue. As the Bunpro lesson points out, it is a causal equivalent to けれども or even more formally が. There are other Japanese words that could translate as “but” in English - the Japanese hint for this grammar point includes one of them, しかし - so when you’re dealing with a fill-in-the-blank question like this where there are multiple potential options the grammar point hints will help you narrow it down to the desired expression.
You should follow the pattern of talking through each example sentence like this, so you clearly understand what function けど is playing in each one. Don’t treat each grammar point as isolated, but actively make and follow connections between points as well as being the expression and the rest of the sentence. Studying grammar is about creating a mental map of the way the language operates; just as roads connect towns on the map, you must establish a web of connections that show how all the different parts of the language work together.
I’d recommend availing yourself of the links to other online explanations of these grammar points that Bunpro includes in each grammar point. Seeing explanations from alternative angles might help you get a better grasp of what’s going on, and the more exposure you can get to different examples of the point in context, the better. If you’re not familiar with grammatical terms in English, it might be worth doing some study of English grammar. Even though the English and Japanese grammar systems are very different, knowing how grammar operates in a language you’re already familiar with will help you know how to approach grammar in a different language.