The main damage that a bushfire can cause that makes a drought worse is the lack of vegetation.

A bushfire could make a drought last longer

When vegetation, like trees, burn, they no longer are able to release any moisture into the air. The less moisture in the air means that clouds are less likely to form. With less clouds in an area, particularly in a catchment area a drought could be prolonged.

When a fire is in progress it reduces the humidity in the air also. This can sometimes reduce the chance of rain. The wind can also move humidity away from an area for example out to sea.

A bushfire can cause damage to water catchment areas

Also when a bushfire burns an area that is in a water catchment area it can contaminate the water supply, for example the run off can get thick with ash. This can make the water less useful for a dam that runs into it.

For example the bushfires in Victoria in early 2009 caused some catchment areas to be damaged.

" Around 30% of Melbourne's catchments have had some degree of bushfire damage. Most of this is in the O'Shannassy and Maroondah catchments."

reference: Melbourne Water - Bushfires and Catchments

why drought and fires have a connection

The most obvious connection is that when there is less rain, the plant materials tend to dry out and even die. This allows for an increase in fuel loads. The bigger the fuel load, the larger the fires.

When an environment or area has less vegetation greater winds can occur. If there are less trees for example the winds can keep going in a straight strong direction. The stronger the ground winds might make a fire even worse.

Another possibility is that when rain only occurs occasionally the plants might react quickly, build up a lot of foliage(leaves) and then no longer produce more leaves. The leaves that remain can then just flare up again.

If rain is more continuous ( if there is no drought) then there is more moisture in the air and so it is less flammable.

CO2 may contribute to drought

Also, when fires burn , they release large amounts of carbon dioxide(CO2) into the atmosphere. CO2 is a greenhouse gas and so can increase the overall temperature. If this occurs over an extended period of time this may increase overall temperatures. Over a log time this may increase the length of a drought.

Soot, smoke and bushfire aerosols

The soot, aerosols and smoke (smog) produced by fires that go up into the air, can also reflect sunlight back into space. This may have a cooling effect which may also change weather patterns forcing some areas to receive more rain than others.