How are fats and proteins broken down in cellular respiration?

Fats and proteins are first broken down, like polysaccharides, into smaller molecules. They then must be modified accordingly before slipping into a step of glycolysis, the transition stage, or the citric acid cycle (if it is a protein).
Fats must first be digested into smaller molecules; namely glycerol and fatty acids. The glycerol is converted into glyceraldehyde-3-phosphate, an intermediary of glycolysis, and added right before the energy payoff phase (where NADH and ATP are beginning to be produced).

Fatty acids, where most of the energy from fats are stored, are broken down into two-carbon fragments by a metabolic sequence called beta oxidation. These two-carbon chunks are then added into the citric acid cycle as acetyl Coenzyme A (or acetyl CoA). Interestingly enough, NADH and FADH2 are also generated during beta oxidation and are useable by the electron transport chain to make even more ATP, so the two-carbon chunks are making ATP even before they enter the cycle!

Fats are excellent long term energy storages due to their chemical structures and comparatively higher energy levels of electrons (compared to carbohydrates). They are chock full of energy - a single gram of fat oxidized by cellular respiration will produce more than twice as much ATP as a gram of carbohydrate. It's not necessarily good news for us - that means an individual must work harder to lose a gram of fat rather than carbohydrate because there are so many more calories in each gram.

Proteins on the other hand must be digested into their monomers, amino acids. When the amino acids are present in excess (amino acids are also used to make new proteins), they may be converted by enzymes into intermediates for cellular respiration. Before they can enter, they must have their amino groups removed through a process called deamination. The nitrogenous leftovers are excreted by the organism in the form of ammonia, urea, or other waste products. The once-amino acids are now able to enter glycolysis, the transition stage, or the citric acid cycle.