I am a Canon user, so I can properly advise on Canon options.
Good and affordable APS-C dSLR option for bird photography is the Canon 7D2 or less expensive 80D and Canon EF 400 f/5.6L or EF 100-400 f/4.5-5.6L IS I or II.
Cameras: 7D2 is pro-grade-construction "bombproof", heavier, more water resistant, has more autofocus points and has good autofocus customization options, has fixed LCD, has "joystick" control as well as top wheel and back wheel for changing parameters, has more custom programming settings, has maximum 10 frames/ second continuous burst shooting. I own this camera. I formerly used the Canon 60D, which is a capable old (2010) APS-C dSLR camera with much less sophisticated autofocus capacity than either the 2010 7D original or the two 2015-2016 cameras 7D2 and 80D.
80D: more plasticky construction, lighter, less water resistant, burst mode 7 per second, fewer (but respectable number of) autofocus points, has movable LCD for video shooting
Both cameras have the ability to focus at the center point at f/8 maximum effective aperture given appropriate lens and teleconverter combinations (there are tables showing compatibility).
Canon EF 400 f/5.6L is a 7 element lens, very well balanced, very light (1.1 kilo), very sharp, very fast AF, and very affordable new or used, because it is an old and relatively simple design. Once one has practiced enough to get the hang of smooth panning and shutter release, the lens is a joy to use for hand-held bird-in-flight photography. Minuses: no stabilization means that for stationary (perched) birds, one needs either a fast shutter speed or some support (tripod or monopod); relatively long minimum focus distance. I have used this lens for 7 years.
Canon EF 100-400 f/4.5-5.6 L IS II. More versatile due to zoom and due to stabilization, sharp at all focal lengths (equivalent to affordable prime lens quality), somewhat heavier (1.6 kilo), packs better for air travel (ie shorter when at 100mm), has much closer minimum focus distance, enough so to get 1:3 magnification at 400mm. Also more expensive. I recently bought this lens. I am still getting used to the different balance and slightly increased weight of the lens.
Canon EF 100-400 f/4.5-5.6L IS I: Push-pull ("trombone") zoom, which some people love and some people hate. The new version has "twist" (usual movement) zoom. Early generation stabilization, less effective than version II. Slightly less sharp than either the 400 f/5.6L or the 100-400 v. II, but I have seen many really good photos taken with "the trombone" at 400 mm. These lenses are no longer made (replaced by v. II), but are found on the used market, and are very affordable.
Appropriate 1.4x teleconverter: Canon 1.4 TC v. II works for EF 400 f/5.6 and version 1 EF 100-400. Canon 1.4 TC v. III works for those lenses plus the newer version 2 EF 100-400.
Yes. good gear helps, and reach is essential, but practice helps more. Field skills help more. What habitat does that species of bird use, where is it likely to feed and drink - very useful to pick most likely site to find said bird species. Bird temperament - will you need a blind/hide because the species tends to be jumpy and shy, or can you just sit/stand quietly in one place, allow the bird to see you and figure out that you aren't a threat, and wait for the bird to come closer. Join the local Audubon Society, join any local nature photography club, check out eBird and local internet resources to get info on sightings.
Finally, read this excellent site for how-to and for useful tips:
http://www.digitalbirdphotography.com/ And Glenn Bartley's site has good advice and great photos.
Peter Ait mentioned the 1" sensor Sony bridge camera RX10 IV, with 25-600mm equivalent lens. This is the hot item for bird photography now, because the autofocus is said to be capable of capturing birds in flight reliably. I have not used it, but I have seen good reviews of it for birding, and there's no question that this is one of the simplest and least expensive options, and is lighter weight than the APS-C format dSLRs and appropriate lens. I would love to try it - it would be a great travel and hiking camera for birding.