Questions regarding DSGE model

What does "competing forces in the economy" mean?

That is not a precise economic jargon just a general statement.

By reading the context of the passage Christiano et al (2017) just generally describes the fact that economy is full of trade-offs. Workers that can be used to build cars can be used to build boats. Worker cannot simultaneously at the same time be building a car or a boat. Thus carmakers and bootmakers compete for workers via market mechanism where they try to outbid each other and where also workers compete each other. All these actions are 'economic forces' that affect the state and development of an economy.

Why we can do experiments is in dynamic stochastic general equilibrium (DSGE) models?

Because dynamic stochastic general equilibrium (DSGE) models are simulations of an economy, and simulations always allow you to do some (simulated) experiments. Experiment is just rigorous examination of some hypothesis e.g. hypothesis claiming some policy or factor influences economy in this or that direction when you manipulate just the factor of interest and other factors are held constant.

For example, in physics it is possible to build a simulation of a star in Fortran, by specifying mathematically all the physical forces present and the composition of a star and so on.

Once you create such simulation of a sun in a computer it is trivial to conduct experiments. In computer you can just easily create a 'clone star' where you keep everything else equal save for one parameter like for example you decide you increase metallicity of a star and see what happens in the simulation to factors like star's brightness or whatever you are interested in.

In the same way as in the analogy above, DSGE model is nothing else just a computer simulation of an economy. Once you simulate economy in a computer it is trivial to run an experiment that tells you what would happen if you would want to implement some policy like for example increasing tax rate of a country. A good introductory text to how this is done is Wickens Macroeconomic Theory A dynamic General Equilibrium Approach.

However, an important caveat is that any experiments done within a simulation are only as good as simulation itself. No simulation is perfect, not even the ones done in physics. Hence you should always keep this caveat in the mind and not necessarily assume simulated experiment provides as good or robust evidence as actual experiment.