The low $l$ power in the CMB spectrum is poorly defined because of cosmic variance - essentially there are a limited number of "samples" that can be used to characterise the temperature variance on large angular scales. The state-of-the-art measurements from Planck (see below) show that the slope of this region is actually undefined. That most experiments agree with a general upturn at low $l<10$ does not give it any greater significance, since all experiments are measuring the same universe. If anything, there has been a greater focus on the $l=2,3$ power being lower than at larger $l$ values - (see here).
The CMB power spectrum is normally presented with the first two terms $l=0,1$, removed, since $l=0$ represents the average temperature of the CMB and $l=1$ is inextricably confused with the observer's motion with respect to the CMB (the large-scale dipole anisotropy).
As far as I can tell (see Ade et al. 2015b), the CMB power spectra that are normally shown have been cleaned of the monopole and dipole contributions, as well as zodiacal light, foreground emission and contributions of the Sunyaev-Z'eldovich effect from nearby large galaxy clusters. Cleaning routines do not remove the Sachs-Wolfe or integrated Sachs-Wolfe contributions that are diagnostic of cosmological parameters and which are considered an intrinsic part of the CMB.
The power spectrum components with $l < 100$ come from regions of space that were not in causal contact at the time of recombination (the so-called horizon problem). This is a "problem" because it is then hard to work out why these causally disconnected regions have such similar temperatures.
The proposed solution is cosmic inflation. In this model the universe expanded exponentially in the first fraction of a second, meaning that the entire observable universe was in causal contact prior to the inflationary epoch, achieved homogeneity and isotropy; but then inflation grew smaller regions to the extent that they departed from causal contact by the end of inflation.
If this were true, then the low $l$ components of the CMB are partly a fossil of the inflationary epoch and thus a probe of the "initial conditions" of the universe, "unprocessed" by what happened afterwards. The variations in the observed CMB at these large scales are due to density perturbations at recombination that lead to small variations in gravitational redshift and time dilation which lead to corresponding temperature variations - the Sachs-Wolfe plateau.
One of the main predictions of most inflationary scenarios is that the power spectrum of these fluctuations should be almost flat at large angular scales - that is, the plotted power spectrum amplitude should go as
$$ C_l l(l+1) \simeq A l^{n-1},$$
where $n \simeq 1$ is the index that determines the spatial fluctuations $P(k) \propto (k/k_0)^{n-1}$. Thus modelling the low $l$ power spectrum could give an amplitude and a "tilt" produced by inflation. In slow-roll inflationary models, the tilt and amplitude of the Sachs-Wolfe plateau are directly linked to the form of the scalar potential that drives inflation. To produce a "red" slope with $n<1$ one could adapt particular forms (e.g. monomial potentials with $V(\phi) \propto \phi^{p}$, $p>0$), but even these do not predict values of $n$ much less than 1 (see Visinelli 2016). In practice, the current best estimate is $n = 0.965 \pm 0.005$, largely constrained by Planck data with $l>30$ and the polarisation signal at low $l$ (rather than the amplitude power spectrum, see Ade et al. 2015a, but still dominated by $l>10$ frequencies). Thus this may be inconsistent with "scale-free vanilla inflation" models, but seem consistent with "slow-roll" inflation.
However, as the CMB propagates across the universe it gets modified by the gravitational structures that it encounters. A photon falling into a potential well is gravitationally blue-shifted and then redshifted again as it climbs out of the well. In an expanding universe this process is not symmetric - there is a term introduced by the time variation of gravitational potential integrated along the line of sight, known as the integrated Sachs-Wolfe effect (also includes the Rees-Sciama effect in the non-linear regime) and is strongly dependent on the amount of dark energy in the universe.
The integrated Sachs-Wolfe effect is predicted to produce a small upturn in the power spectrum towards the smallest $l$ values ($l<10$, see the picture below from an article by de Putter et al. (2010). This cannot be clearly seen in the raw measurements of the CMB power spectrum because of the large cosmic variance uncertainties. The reality of the effect has been confirmed however (even with WMAP measurements) by cross-correlating the CMB map with maps that trace the large scale structure of the universe (e.g. big galaxy and cluster surveys - see table 1 of Dupe et al. 2010, which gives a review of methods for detection of the effect).
In summary, the overall picture is messy. The form of the CMB spectrum at low $l<10$ is affected to some extent by the parameters of an inflationary model (if inflation is a correct prediction of what is going on), but is more affected
by the integrated Sachs Wolfe effect, which in turn depends on the equation of state of dark energy (as a function of time!).
