RS485 without flow control
RS485 is a multidrop bus, which means that a number of nodes are connected to the same bus and, in this case, they all have the potential to "talk" at the same time. For a device to take control of the bus and be able to talk, the RS485 transceiver (in your case the MAX3160) has to have its transmit enable pin be enabled. All the other devices on the bus need to have them disabled. If more than one device is enabled to transmit, data can become garbled because one device might be trying to drive the bus in one state and another device trying to drive the bus in a different state. 

You enable the device to take control of the bus by setting the DE pin high. Once it is high, anything on the DI pin is transmitted on to the bus. It is important to note that all other devices on the bus should have their DE pins low. 

The other enable pin on the transceiver is usually called the ~RE pin. Note that this is low for anything on the bus to be reflected on the RO pin. That looks like it's the R1OUT on your Max device. 

A lot of the time, you see the ~RE and DE pins connected together and to a GPIO on the micro. This GPIO determines whether you are listening to what other nodes on the bus are saying or talking on the bus yourself. 

All this is not to be confused with flow control which is often used on devices such as modems that are relatively dumb and have relatively small data buffers. This is a method of each device saying it is ready to send more data out or ready to receive data in. This can be done in hardware with RTS/CTS pins and can also be done in software with XON/OFF messages. There are lots of good resources on line that explain this very well and I still dip in and out of these every now and again because it can get confusing! 

It's probably worth noting that there are various methods of deciding who is allowed to talk on the bus at any one time. There always needs to be a method of knowing who is the one master at any one moment in time. 

So, to listen on the bus, ensure ~RE (R1OUT) and DE are low. To transmit on to the bus, DE should be high. If ~RE is high at the same time, you will not get what is being transmitted, send back to the Rx pin of the UART on your micro.