Why do we say
el cepillo del pelo
el cepillo de dientes
el cepillo de los dientes
Cepillo de dientes is a recognizable, often-used expression that points to a specific kind of object that is also used often, so it works as a fixed phrase. It's the kind of phrase that in a language like English or German would tend to form a single-word compound. In Spanish cepillo de dientes forms a three-word nominal phrase that goes together as if it were a single noun, and you cannot change its parts (e.g. you cannot say cepillo de los dientes or cepillo para dientes; it would be understood but sounds very odd). You also have pasta de dientes ("toothpaste"), which is also a fixed phrase, and these two in a way reinforce each other.
The thing one uses for brushing one's hair is called cepillo alone in most cases, just as the thing one uses for brushing clothes or fabric or the fur of one's pet. Cepillo is a very general term and it covers all that. You add a specifier like de la ropa or para el pelo only when the context doesn't make it clear, and these specifiers work like normal Spanish phrases: you can change them a bit, since they don't form fixed phrases.
In the end there's no reason, as such, why some expressions are fixed in a way and not in another, and some other expressions are freer; you just have to learn them as they come.
Unlike English, the use of articles or prepositions in Spanish do not bear any resemblance — it may in some cases, but this is not a rule; we do, nevertheless, use set phrases when it comes to article or preposition placing.
As stated by Pablo's answer, you'll have to learn these set expressions as part of phrases; although, if it's of any consolation, this is the same problem with which we, as non-native speakers, have to deal as part of our English language learning.
If you use 'de', it means what comes after it is specifically connected to the function of the noun, as opposed to 'del' or 'de la'. It is connected to a specific quality of the noun.
If you say mi casa de playa, you mean your beach house, which is not a casa de la playa, which would imply the beach has a house.
For example: la ropa de la chica, the clothes that belong to the girl (some girl, in a conversation); whereas ropa de chica, girls' clothes, as opposed to boys' clothes.
This is just an explanation for a beginner.