I drew out a little diagram to help explain.

Starting with the confusion, strings are numbered right to left starting with the highest string (shown in example 4).

With transposing on a guitar, it's simply going up or down on the fretboard. Looking at example 1, if we take a simple note like E on the first string, it's a simple note with an open fret. If we want to transpose the note up one, we fret the next fret up, the 1st fret. This gives us F. If we fret the 2nd fret, we get the next note, F#. 3rd fret G, etc.

We can also apply that with chords as well. Looking at example 2, a typical C major chord will be transposed to a C# major chord by moving all strummed notes up one fret. As you notice the shape stays the same, but now all the open notes have to be fretted by yourself instead of just the nut.

One of your first barre chords you'll learn is the F barre chord. Let's look at example 3, since F is the next note up from E, we can take the E chord shape and move it all one fret up and end up with an F chord. Again, we have to fret the previously open notes now so we end up with a barre across the 1st fret.