Please have a look at this forecast paper: 1611.00036. I will cite some important figures.
There are two important observables in the large-scale structures: Baryon Acoustic Oscillations(BAO) and Redshift-space Distortions(RSD). BAO tells us the expansion history of the universe, so measuring BAO puts constraints on the equation of state(EoS) of dark energy.
For example, if the EoS is parameterized like $w=w_0+w_a(1-a)$. DESI expects to see such a contour plot($68\%$ limit) if the "true" value is $w_0=-1$ and $w_a=0$:
Here "P" means Planck prior and "BB" means the case when board band information (the shape of measured density power spectrum) is used. Blue line is previous result obtained by BOSS project, so we see huge improvements.
Similarly, RSD tells us how structures are clustered and grow, while different gravity theories predict different growth rate. Therefore DESI will give stronger constraints on gravity theory (modified gravity is an alternative explanation of the acceleration of the universe):

And of course DESI will have better measurement of the total amount of dark energy:

update:
It is a long story from raw data to cosmological parameters, but the procedure can be briefly but not precisely summarized:
The raw data is the positions (RA, DEC, redshift) of many galaxies/quasars. These positions will be transformed into statistics, e.g. power spectrum and correlation function.
And then people adopt one kind of parameterization/model to establish the likelihood $\mathcal{L}(d|\boldsymbol{\theta})$, here $d$ means data and $\boldsymbol{\theta}$ means parameters.
With the aid of Bayes theorem $p(\boldsymbol{\theta}|d)\propto\mathcal{L}(d|\boldsymbol{\theta})p_{\mathrm{prior}}(\boldsymbol{\theta})$, we will obtain the constraints/measurements on these parameters.