5.131 If the truth of one proposition follows from the truth of others, this finds expression in relations in which the forms of the propositions stand to one another: nor is it necessary for us to set up these relations between them, by combining them with one another in a single proposition; on the contrary, the relations are internal, and their existence is an immediate result of the existence of the propositions.

Here W. seems to confuse statements and their referents (as is often the case when the term "proposition" is used).

In general terms - not W.'s - an interpretation of a statement comes about through some function that assigns set-theoretical entities to the parts of the statement.

In the case at hand, such as - to take a fairly specific example - "(x)(A(x)) |- (Ex)(A(x))", in words: if every thing is A, it follows that some thing is A", we might use a function i() with the properties that
i( (x)(A(x)) ) = i(A) = some set and i( (Ex)(A(x)) ) inc i(A) & ~ i ( (Ex)(A(x)) )=0.

In words, this may be summarized as: i() assigns to the expression "(x)(A(x))" a set i(A) and to the expression "(Ex)(A(x))" a non-empty subset of i(A). Now if we also stipulate that "p |- q" iff "i(q) inc i(p)", i.e. q follows from p iff the set of things that serves as the interpretation of q is included in the set of things that serves as the interpretation of p, and if one stipulates that i(  (x)(A(x)) ) is to non-empty sets exclusively, one has - in principle, and with a considerable amount of detail left unstated - a tolerably clear explanation of "follows from" in the logical sense, at least in the case of the present example, of explaining why "some things are A" would "follow from" "every thing is A". (There are other ways, but I wanted to avoid discussing the semantics of variables.)

To recapitulate: "some things are A" does "follow from" "every thing is A" if we use "every thing is A" to refer to A's indiscriminately, and we use "some things are A" to refer to A's discriminately (and reconstruct this in terms of set-theory, where one uses a function to map parts of speech to sets of real (and possible) things).

But this type of explanation - that owes most to Tarski - is not what W. had in mind. It may serve to show what W. confused: The relation between the statements "p" and "q", such that e.g. "q" contains only variables also contained in "p", may or may not be somehow be correlated to the relation between the (sets of) things the statements "p" and "q" represent.