Tab into edit mode
Enable the 'magnet icon' which is the snapping tool.This is used to snap an edge,vertex,face,volume etc.
Next, enable the other icon which has 'two dots in the corner and a line connecting it'.This is the 'Auto merge' option.It is very useful and it automatically merges overlapping faces vertices or edges. But Auto merge doesn't remove internal faces
Now with all these tools enabled select your face and press 'E' to extrude.
Hover over the edge in which you want the extrusion to stop. Now left click to finish.
And viola!! you have filled your mesh!
The tools used here:(just in case you have trouble understanding)
Name: Snapping/Snap tool
Object, Edit, and Pose Mode
The ability to snap objects and mesh elements to various types of scene elements during a transformation is available by toggling the magnet icon in the 3D Viewport’s header buttons.
Snap to vertices of mesh objects.
Snap to grid points. When in Orthographic view, the snapping increment changes depending on zoom level.
Snap to edges of mesh objects.
Snaps to the surfaces of faces in mesh objects; This is useful for retopologizing.
Snaps to regions within the volume of the first object found below the mouse cursor. Unlike the other options, this one controls the depth (i.e. Z coordinates in current view space) of the transformed element. By toggling the button that appears to the right of the snap target menu (see below), target objects will be considered as a whole when determining the volume center.
Snaps to the middle of an edge. This snap element only pertains to mesh objects.
Snaps to the nearest vertex in an edge that makes a perpendicular angle. This snap element only pertains to mesh objects.
Multiple snapping modes can be enabled at once by Shift-LMB the different snapping elements.
Mode: Edit Mode
Menu: Sidebar ‣ Options ‣ Tool ‣ Auto Merge
When the Auto Merge option is enabled, as soon as a vertex moves closer to another one than the Threshold setting, they are automatically merged. This option affects interactive operations only. If the exact spot where a vertex is moved contains more than one vertex, then the merge will be performed between the moved vertex and one of those.