Michael Dummett's The Logical Basis of Metaphysics should be a difficult book to read since it challenges the common sense of many people. It may be difficult to see what his point is, or to take that point seriously, given that common sense. On the other hand Dummett is an analytic philosopher. His attempts at precision are not intended to confuse but to lead to clarity.
One approach to the book is to read the introduction, "Metaphysical Disputes over Realism", and then the last chapter, "Realism and the Theory of Meaning". The introduction will describe realism in terms of the principle of bivalence and the last chapter will illustrate various ways an anti-realist might respond. Using that as a framework, one can read (and re-read) the entire book.
First let's consider the introduction.
Dummett takes metaphysical topics such as the soul, free will, God, ethics or the future seriously. Up front, in spite of the "destructive phase" of analytic philosophy that may have treated such topics as meaningless, he asserts: (page 1)
...if philosophy does not aim at answering such questions, it is worthless.
He just stepped on the common sense of anyone still enchanted by logical positivism.
Since he is an analytic philosopher he will approach metaphysics through language and logic. To understand these problems better he breaks metaphysical topics into classes of statements and notes that people who take a particular class of statements as representing something real also take a particular logical attitude toward those statements: (page 9)
It is difficult to avoid noticing that a common characteristic of realist doctrines is an insistence on the principle of bivalence - that every proposition, of the kind under dispute, is determinately either true or false.
He just stepped on the common sense of those who only accept classical logic.
Now let's look at the last chapter.
A person who accepts realism does so only with respect to a particular class of statements. If some person accepts the realism of Cantor's transfinite numbers, that does not mean that the person also accepts the realism of statements about the future. Each dispute is over a specific class of metaphysical statements. (page 322)
...these are all disputes concerning the correctness of a realistic interpretation of some class of statements, which we may call 'the disputed class'.
While realism is split between different classes of statements, there may be many anti-realist responses for the realist position supporting bivalence for statements in a particular disputed class. Anti-realism is not a unified position against the realist. This chapter attempts to show the different ways one can be an anti-realist although generally the anti-realist interprets the disputed class, not realistically, but in terms of other statements called "the reductive class" (page 322). These can be so interpreted in a "weak form of reductionism" or a "full-blooded reductionism" (page 323).
This steps on the common sense of those who think of oppositions as unified positions. These disputes over metaphysics are not disputes between a unified realist position opposed by a unified anti-realist position.
Dummett admits in the preface that he has not completed his task: (page x-xi)
My aim was to achieve a prolegomena to the work I still hoped to do in philosophy, and regard as one of its major tasks, to resolve the problems concerning realism in its various specific manifestations. I have not yet made substantial progress with this task, and now probably never shall; I shall be content if I have persuaded sufficiently many people of its importance, and of the correctness of my strategy for tackling it, to make it likely that others will achieve what I once hoped to.
Hopefully the above provides the context the OP is looking for to begin reading Dummett's work.
Dummett, M. The logical basis of metaphysics. (1991) Harvard University Press