In French, why is there a "du" or a "de" before certain words?
I really hope this isn't a stupid question; if so, I apologize for plaguing the forums with it. Onto the "question". I've noticed that some words have a "du" or a "de" before them, such as: du bœuf, du beurre, and du pain. Why is that? I appreciate all answers.
These are partitive articles so they denote SOME unknown quantity being consumed. Du is a contraction of de + le. No such thing as stupid questions, I learned from this by researching it: https://en.wikibooks.org/wiki/French/Grammar/Articles
What you're describing is the phenomenon of partitive articles.
This type of article is used to indicate that the noun in question is of an unspecific quantity; the English equivalents of this are either "some" or not using an article at all. You could say, for example, either "I'm drinking some juice" or just "I'm drinking juice".
The latter isn't acceptable in French, which in most instances  demands that every noun be preceded by some sort of determiner (a broad grammatical category that includes, among other things, articles). You must therefore add in some sort of determiner, of which partitive articles are the best option; the partitive article communicates that you don't know or don't care to specify how much juice is being drunk.
The use of partitive articles is not restricted to food, but they usually find vigorous use when describing food just by virtue of their function.
The partitive articles are formed with de + the definite article. de forms contractions with le and les - not specifically when forming the partitive article, but they do come into play in such a case:
Masculine singular: de + le → du
Feminine singular: de + la
Unisex singular before vowel sound: de + l'
Unisex plural: de + les → des
^1 In the majority of cases. This rules applies enough that you're better off memorizing the exceptions rather than the situations that it does apply in.
Firstly, I believe that there is no such thing as a stupid question if asked with good intentions, so you don't have to apologise :) About your question, du is some (masculine) and de is some (feminine). If you notice that when you have a question with one of these on Duolingo e.g.
Il mange du pain
When you type in the answer, you can either say "He is eating some bread" or "He is eating bread" and either one will be marked correct. Based on this, I'm pretty sure that 'du' or 'de' are used generally to stick before certain nouns instead of saying "He is eating THE bread" where you are referring to some specific loaf of bread, rather than bread in general. About what nouns this applies for, I really don't know, it seems that they all might be food objects. Also, I don't know if what I said is true, so please correct me if wrong :)
Oh, my apologies, I should have clarified that better. I meant why is it there at all. From my understanding, du or de means "some" or "of", correct? So if I said "Je suis mange pain avec beurre" instead of "Je suis mange du pain avec du beurre", what difference does it make?
The difference is that "du pain avec du beurre" is correct and "pain avec beurre" is not.
French uses articles much more often than English, in fact words are almost always preceded by some kind of article or preposition. Where in English, using "some" before bread and butter is optional, in French you cannot drop the "du" because you need an article.
Also as demilade1410 pointed out, "je suis mange" is incorrect, as "je mange" can mean both "I eat" and "I am eating"