The spark arrestor is a small screen that keeps sparks emitted by the engine from exiting the muffler. Over time, the arrestor can become clogged with soot which may cause the chainsaw engine to stall. You can try cleaning the arrestor with a wire brush but you may need to replace it with a new one.
The residue that results from old fuel being left in the tank can create a restriction in the carburetor. This is why you should always use fresh fuel when filling the tank and a fuel stabilizer to help maintain the quality of the fuel. For better stability, you can use a pre-mixed fuel and oil product that is ethanol free. You can try cleaning the carburetor with carburetor cleaner to eliminate the clog. If cleaning is not effective, you can often purchase a carburetor repair kit to replace some of the components or install a new carburetor altogether.