How can I repair a dining room table with folding ends?
Looking at the picture it seems that there may be several problems going on here. In the foreground the swing out arm seems to have come completely loose from the vertical support between the legs. The C-Clamp being used to temporarily hold that part in place is not coming close to holding the hinged piece in place so it can work properly. For this side the assembly needs to be properly re-glued or screwed in place. A combination of gluing and hidden screws from the inside of the vertical support may do the trick.

For the support on the far side of the picture it looks like the top portion of the swing out arm has completely split off. Fixing this part will require replacement of the arm part or as a minimum squaring off the split top face and gluing a new piece to replace the broken away part.

One additional note is to evaluate the condition of the vertical support member between the legs. If this piece is wobbly and not secure then I would suggest that the repairs are much more involved and you may be faced with making a decision if the table is worth the effort repair at all. Overall woodworking experience and / or willingness to take on something like this may play a big role in the decision making process as well. 

Evaluation would need to be made for the other end of the table to see if it has similar problems or not. 

If it was me and the table had some sentimental value I would dive into the repairs. Doing repairs like this however will not do a lot to preserving any heirloom quality of the table but that has probably been lost a long time ago as noted from current condition.

I have one final comment to make. When trying to wedge up the swing up portion of the table top it is better to place the wedge out at the end of the arm as opposed to near the swing up hinge. As shown in the far side of the picture the  wedge stick places an undue amount of stress on the hinge joint that could lead to further failure.