In this excellent polyphonic study, the performer must attempt to keep all four lines in mind.
Similar polyphonic difficulties can be seen in Rondo movement of the Sonata in G Major, Op. 31, No. 1. The fingerings provided in the score may be tricky, and include finger exchanges, and unusual crossings. In spite of the difficulty, however, the given fingering should help you to hold all notes for their full indicated value, and to maintain the continuity of the lines. The pedal may at times help, but be careful that it doesn’t make the lines muddy and unclear. The melody is of primary importance, but observe the chromatic and diatonic stepwise movement in the lower voices. A good suggestion for understanding these multiple lines would be to play through each one individually several times from beginning to end. After that, use both hands to play the two lines of the right hand, then the two lines of the left, observing and listening to the way they interact together. Once all parts are played together after this preparation, the lines will be much better understood. You may yet find a place or two where a certain note might be better played by the opposite hand indicated. An example is measure 8, on the final beat. It may be easier to play the left hand D with the right hand, and the right hand B with the left. Such note exchanges may not be exactly what Beethoven wrote, but is the better choice if it makes the passage more musical.