I/O expanders main advantage is that you trade 2 wires for 8 digital I/O. You could go up to 64 I/O because you can hook up 8 of these chips to 1 I2C bus and individually strap the A0, A1, A2 pins accordingly. That means that I/O expanders are well suited for like indicators, buttons (if you can take the burden of I2C access), relay boards, etc. You just need a cable with power, I2C SCL, I2C SDA and GND.
Accompany it with a EEPROM/ADC/etc. that also has A0..A2 address pins and you could make some interesting modular systems out of them.
I2C can be multi-master.. so multiple boards could (in theory.. - if you want to figure out the software for that) read or write to 1 chip.
Disadvantages? Well , it's an extra chip which takes up space and price. Sometimes a bigger microcontroller (28 to 40 or 44pin, 44 pin to 64 pin) is just a few ten cents more, depends if you need to upgrade to another family or not. However if want the remote I/O to be on another board, this argument is not valid.
You need to write additional software to talk to the chip, so you can use the extra pins.
You can create much simpler output-only I/O with a 74HC595 or 164 on just 3 pins. These can be operated much faster (MHz territory), and chained much longer.
To even change a single bit, you need to send a complete I2C packet. Given that this chip is 100kHz and you need to write 2 bytes or so to do anything at all - the latency is in the order of 160us - 200us.
As said, for buttons, indicators or relays that's often not a problem, but you may need to buffer port changes together and send them at once.
How to operate: on page 9 there is the I2C address. It's different for a non-A and A version
The next pages show how to read and write. It seems like it basically send address + write output, or send address + read port.
You apparently don’t have to set the direction, with the "quasi directional I/O". Not sure how good or easy that works, but I'm sure it will work fine for just a few LEDs or something.