Looks like a common ol' problem, particularly in trigger shifters which are old and have been sitting for a while. The grease inside the shifter gums up, which prevents the pawls and springs from operating freely, thus not catching when you try to change gear.

On the top of the shifter, you should see 2 or 3 small screws. Undo these, and the top part of the cover and gear indicator should come off. This will expose the guts of the shifter. Get something to flush it out with. I use brake cleaner, as it evaporates, and doesn't leave a residue like degreaser does. Give it a good flush out, while working through the gears. They should start to catch and function properly now. Then, get some light lube like chain oil, or a light grease from an aerosol can, and repeat the procedure. Spray/squirt it into the shifter while working the levers, and your problem should be solved.