To wax lyrical means to be effusive and enthusiastic about something or someone and dates from the latter half of the 19th century. The word ‘wax’ used in this sense means to increase in size and can be traced back to Anglo-Saxon times. It is the opposite of wane. It is largely obsolete today except in the waxing and waning of the moon and in expressions like wax lyrical. Wax lyrical is probably the most familiar, but there is no reason why people should not ‘wax poetic’ or ‘wax philosophical’ if they so wish.