This advice is an educated guess.
I marked up one of your photos
Currently the spring has no tension on it because the tension rod is not hooked on anything.
The hex head in the yellow circle may be to put a socket on so you can rotate the spring tension plate where one end of the spring is currently hooked, thus pulling on the spring and creating tension in it. The spring tension plate appears to rotate about the red circle.
If the hex head in the yellow circle can be rotated with a wrench and as you rotate it you are pulling the spring, then you are going in the right direction.
With tension in the spring, find where the mushroomed head on the end of the tension rod hooks to such as the rear axle. The spring now will have tension and keep the tension rod from falling off and at the same time apply pressure on the pulley keeping the belt tight.
I think the belt is on correctly it just that the spring has no force in it to keep the pulley pressed against the belt.
Dave M7040