The primary winding rating tells you (effectively) the compatibility for mains power in your region, so make sure a replacement is also rated for 120VAC and 60Hz.
The secondary winding rating tells you most of what you need to know to size the power supply. In this case it's a 5:1 step-down transformer (changing 120 to 24 volts), then 20VA means it supplies up to 20 watts of "apparent" power. (The discussion of real versus apparent power (VA versus watts) can get rather technical, because it depends on the type of device you are powering.)
You should find a replacement that provides at least the same minimum power (20VA). If you treat this as 20 watts, you can calculate (\$P=IE\$) that the secondary side is providing about 0.83 amperes. It doesn't hurt to have extra current capacity, so if you find a 24VAC 1A or 24VAC 2A supply, it will work.
Finding a power supply with the correct socket to match the cable you have might be more challenging. The connectors are usually referred to as a "2.1mm DC barrel jack" style (even though your transformer outputs AC). Hopefully that will provide you with some specs and keywords to find what you need.