I am going to try to be as clear as possible to you, even though I am unclear what the question exactly is that you are asking. I would like to point out first that this is a really hard theory and that you are going to have to put in a lot of effort to get this. There are many philosophy student who do not understand this, even after finishing their study. Remember that.

I think what you are asking is what synthetic a priori judgements are. I do not know the details of your question because you are using your terms in a "loose" manner, based on language conventions which you use in your daily life (and I am not part of that). The parts I am unclear about are:

1) I pretty much have to describe Kant's basic strategy of securing
2) I understand that he believes his transcendental idealism is the solution to the problem,

Note: What is in prolegomena is also in "Kritik der reinen Vernunft". Perhaps that can help you along. It does take a while to read, and, as I said, it is a very hard philosophy.

Anyway, to get to the point:

Synthetic a priori judgements are judgements which have added something to a perception by the means of a priori intu?tion.

Synthetic judgements are in a way the opposite of analytic judgements. Analytic judgements are judgements which are made within the limits of what is known. If, for instance, someone had determined that for plant p to grow three things are necessary: dirt (d), water (w) and sunlight (l), I would be able to analyticly conclude that if p would not grow, but wither, and d and l are in abundant supply w is absent. Synthetic is the opposite of that in the manner that we look at the given facts, but do not know the answer yet and so cannot pick it out of an available set of answers. If we would know what 7, 5 and + means we could conclude synthetically that 5+7=12. This without knowing 12 on forehand.

A priori is perhaps best understood when contrasted with a posteriori. A posteriori means after perception and a priori means before perception. Kant speaks of this in the context of brain functions. Kant is what people would call a rationalist, which means that he believes that to understand our perceptions we need to know how to deal with them a priori. Empiricists believe that we percieve and on the bases of our perceptions we learn to think. Rationalists give empiricism a place a posteriori. In a rationalist view one knows what one percieves (in a basal way) a priori, and works out details (reasons) a posteriori.
A priori are what Kant calls "intuitions"; which he has divided into categoria. All categoria are based on a basal understanding of space and time. The categoria break down into four groups: Quality, Quantity, Relation and Modality. I would, for instance be able to say a priori that any object "has" these four categoria. When someone speak of a door for example I can a priori say that it has a certain size, that it is open, cloed or inoperative, etc.

Combining the two:
Synthetic a priori is per the definition of synthetic after perception and per the defintion of a priori concluded by means of a priori intuitions. If I would for instance be discussion geometry I would be able to work out how to draw the third line of a triangle because I percieve two lines, know that a triangle has three corners and have a basic intu?tion on space and time (in this example space (2 dimensional (hopefully) ) is the a priori intuition). I needed the perception of the two lines and the information that it was supposed to be a triangle for my a priori intuition to point to an incomplete picture in the sense of "extension" so my brain could conclude a line was "needed".

I hope this makes it any easier. If there are still unclarities, just ask.

Here is great straight forward resource:

Kant: Judgments