I'm very new to radio astronomy and I am curious how an the signals received by a radio antenna are converted to data on a computer and then images? Is there a popular software to use for this, or do astronomers typically write their own code? Thanks in advance!
There is no universal answer to this question. Instrumentation, data formats and software differ greatly between telescopes.
Most software packages in use are written by astronomers as open source and shared and modified widely within the community. But instead of listing popular packages, perhaps a description of the common way of plotting images is a better answer here.
Consider a very basic radio telescope: A dish that's movable in two axis with a simple radiometer (think "power meter for RF") as the receiver. During an observation, you record four-tuples of time, azimuth, elevation and signal strength. Plotting these is pretty straightforward: Convert the instrument coordinates to celestial coordinates, and plot the signal strength over these coordinates. You'll get a classical intensity distribution map (source). Those maps can be created using simple spreadsheet programs like Excel or custom code snippets.
Adding the capability to record spectra to your receiver makes the table you record your data in a bit more complex (since your last column is no longer a scalar value but a spectrum), and allows the creation of intensity-velocity-plots (source). From a software perspective, this is still a simple plot-points-in-2D excercise.
For anything more complex, especially array setups with multiple antennas and/or focal plane arrays, data reduction and plotting requirements are far more involved and highly depend on the actual setup. CASA and AIPS, mentioned in the comments above, are widely used packages, but there is a variety of custom code around.