In order for transitions to work, there must be enough spare material overlapping the join to cover the period of the transition between the adjacent clips.
You can work in two ways with the Premiere timeline: Single track Editing, where all clips and transitions are in a single track (as far as you can see, anyway); and A/B editing, where you can see separate tracks for the two clips and the transition between.
It sound like you are working in Single Track mode.
Shorten the two adjacent clips so that the end of the first clip is before the end of the available footage by a second or two. Do the same with the start of the second clip. (To do this, put the selection tool at the end/beginning of the clip, see the change of cursor, and drag in the obvious direction). Then drag the second clip to butt up against the end of the first. Now when you drop ion the transition, it should extend the ends of the two clips automatically to create a changeover period during which the transition takes effect.
Depending on your hardware and project settings, you may then see a red cross on the output. If so, render by hitting Enter to allow Premiere to calculate the new video frames during the transition.
