You cannot automatically play records through the computer speakers unless you are recording, because the turntable has been designed primarily as a recording turntable. To hear your music in Audacity while recording, make sure software playthrough is enabled on the Audio I/O tab of Preferences, then click in the right-hand recording VU meter with the microphone symbol, which turns on "monitoring". Click in the meter again if you want to turn monitoring off.
If the VU meters are not visible, go to the Interface tab of Preferences and check Enable Meter Toolbar. In Audacity 1.3.3 or later, click View > Toolbars > Show Meter Toolbar.