In theory all training is functional because you always have a reason for doing it- there is always a function. However, when people talk about functional training they are normally referencing training thats main focus is not to build muscle or strength. A good example of this would be Crossfit where training is focused around Olympic lifts but also other exercises found in the Crossfit Games. A much broader example would be training for a sport such as rugby or football. But to find out how to do so you need to analyse what your sport entails.
One example that I have experienced is rugby. Now before I started my various training blocks I thought about what you need for rugby, the physique, attributes and strengths. The main points I came up with were the following- strength,speed and explosiveness. Then I needed to take into account my positional needs as a player, and improvements were made to those original ideas. This is where my personal split came from. Of course because of the personal nature with training what works with some might not work with others.
So now that you have identified the specifics of your sport or positional needs you need to identify what exercises and training volume will be best. If your sport requires speed then Olympic lifts may be best for you, if power is what you are after then follow a power lifting regimen and if a mixture of the two is needed then do that. In regards to training volume it really depends on what time of the year you are training. If you are training in the off season then you can afford to workout more often and with more vigour, but if it is the playing season then you may want to design your training plan around matches and recovery and make sessions a little more relaxed.
Specific exercises are important for functional training as you are relying on them to boost your performance. So you should pick ones that imitate either sport specific movements or ones that have carryover. Another rugby example here but an explosive landmine press is the gym equivalent of a hand off or stiff arm, a push press is close to the movement completed by a lifter in the line out and so on. If you are having trouble with this then please contact me via DM or email for some help.
Often training for a sport does mean avoiding the more attractive things like muscle mass or a low body fat percentage, but it will be for the betterment of you as a player. By doing this type of training you will be helping yourself and those around you in whatever discipline you take part in.