It takes getting used to but the phrasing here 남자가 마시는 is all describing the milk, thus the adjective 차갑다 is as well.
I don't understand how it's describing the milk I thoght it was The man that is drinking milk is cold
The topic is "the milk that (someone) drinks". On grammatical grounds 남자가 "the man" could be the subject either for the verb "to be cold" or for the verb "to drink" (but not both at once), The first option leads to an incoherent sentence: "as for the milk that (someone) drinks, the man is cold". It is incoherent because the statement "the man is cold" seems not to be related to the milk, so how can the milk be the topic? The second possibility ("man" is the subject for the verb "to drink") leads to a coherent sentence: "as for the milk that the man drinks, (it) is cold". This is coherent because the topic, the milk, is also naturally the thing which is cold. So the first option is rejected, not on grammatical grounds, but because of the semantic role the topic, identified by 은/는, must play in any sentence.
I too am having trouble seeing how this is not "The man drinking the milk is cold," as that's there's nothing wrong syntactically or grammatically with that English sentence. So what would it look like in Korean if you said that?
The particle attached to man (가) and milk (는) indicate which is the subject of the verb
So the man is the subject of the verb 'to be cold' or the verb 'to drink'? Is there a rule that a noun tagged by 이/가 must be the subject of the first verb appearing after it in the sentence? Added 2019.02.04: As Ash-Fred says, 남자가 마시는 우유 is the topic of the sentence, where 남자가 마시는 is a subordinate clause that modifies 우유. The trick for English speakers is realizing that in Korean it is possible for a sentence to begin with a subordinate clause. In English the first thing we expect to see in a sentence is the subject, so when we see 남자가 we jump to the (wrong) conclusion that 남자 is the subject of the sentence. There seems to be a word-order rule in Korean that "words of a subordinate clause are contiguous". However the main sentence elements (topic, subject, object, indirect object, verb) are not necessarily contiguous (hence subordinate clauses can be inserted at various places). However this rule does not help us see where subordinate clauses begin. There is also the general rule that old information (e.g. topic) comes first in a sentence and newer information later, with the verb at the end. Also, sometimes the item marked with 는 can be both the topic and function as the subject, as in this sentence. Thus in this sentence, if 남자가 were the sentence subject, rather than part of a subordinate clause modifying the topic, this would be an odd order of things (subject before topic, and different from the topic). Thus it is more natural in view of these rules to think of 남자가 as part of a subordinate clause modifying the topic 우유.
"The milk that the man drinks is cold." has been counted as wrong.... whereas in a similar sentence "eating" was counted as incorrect.
the structure of the sentence is so strange ! man+drinking+milk+is cold become : The milk that the man is drinking is cold. Even that sentence sounds weird to me. The man is drinking cold milk. Would sound better.