I'm presuming you mean handling the thermostat and env monitoring - essentially a DIY incubator, rather than controlling an existing/working incubator? I've not tried it, as we always get our chickens after they've hatched.
Making some fairly general assumptions about what you want to do, I'd look at using a DHT22 to measure air temperature (and humidity). I'd also look at a DS18b20 which are available as waterproof/easy clean versions so you can put it in among the eggs even when they're due to hatch. Having several will help detect hot/cold spots.
For the heating element you just need a motor driver which can source enough current/power. Heating elements can use a lot of power, but the point of an incubator is that you should just need to gently top up the heat, rather than cooking the eggs! A quick search throws up lots of heating elements of 100W+ but they're going to fry your controller and your eggs. 25W is more managable. As long as your driver can handle the power, your can use PWM to reduce the heating rate further. If you've got an element. I'd measure the current drawn from your heating element, and check it against the power rating of available motor drivers.
Personally I'd connect this all up to an Arduino rather than a Pi. A Motor controller is about £2, and the sensors about £1 each (and will plug straight into motor shield's servo pins). The Arduino will not only be cheaper, but as it has no OS/SDcard its more robust for an embedded application like this... drop the power a few times and it will always bounce back. Is your Pi going to reboot, and fire up your code every time? Arduino also has proper PWM, so you can easily set the heating element to fractional power levels - rather than on or off you can turn it on very slightly when the temp falls a little.
For more fancy/high level stuff I'd add ethernet, and use MQTT to push the sensor data out to a broker, and node-red (probably running on a Pi then accessed from my desktop machine). I'd add an i2c LCD text display for monitoring on the actual incubator.