Controlling HDMI Switch via RS232/UART
Your HDMI switcher, and your USB to RS-232 converter speak RS-232.  The Arduino speaks TTL UART.

While the concept behind them, and the overall format of the data, is the same there are major differences:


RS-232 uses a ±10V NRZ signalling method, TTL UART uses a 0-5V signalling method
RS-232 uses inverse logic compared to TTL UART


That means that you need to convert the Arduino's TTL UART to ±10V NRZ with inverse logic in order to communicate with the IOGear.

Fortunately that's a simple enough task using a special interface chip. These are readily available and can be bought for tiny amounts as a breakout board or even a full shield if you like (such as this one from Sparkfun).  The chip is known as a MAX232 in its most common form, though other manufacturers may use different letters.

If you pop on eBay (or your favourite budget seller site) and search for "Arduino MAX232 DB9" (the DB9 is the type of plug on the board - the 9-pin D connector) you will get lots of hits to choose from.