Why in the first Friedmann equation quantity $ρ$ is directly proportional to Hubble's constant despite the fact that gravity counteracts expansion? Here is the first Friedmann equation:
$$H^2 = \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$$
We know that matter and energy through gravity slow down or reverse any expansion in the fabric of spacetime. Yet in some context and specially here with this equation I encounter the fact that matter and energy content of universe is increasing the expansion rate instead of the opposite, as if there's an anti gravitational force in effect. How so?
 A: Your intuition that $\rho$ ought to slow the expansion is correct, but that intuition concerns $\ddot{a}$ not $\dot{a}$ (and indeed in the other Friedman equation you do see a minus sign).
The $(\dot{a})^2$ term comes from the expression for the curvature when the metric is of FLRW form. The Einstein field equation says there has to be a relationship between this curvature and $\rho$, and that is what you are seeing in the first Friedman equation. By comparing with a Newtonian description, it can be roughly interpreted as a statement about energy. $\dot{a}^2$ is related to kinetic energy, and $\rho$ relates to gravitational potential energy.
A: $\dot a$ is the rate of change of the scale factor i.e. it tells us how fast the universe is expanding. Shortly after the Big Bang the universe was very dense and expanding very rapidly so both $\rho$ and $\dot a$ were high. Then as time went by the universe became less dense as the matter was diluted by the expansion, and at the same time the expansion slowed as the gravitational attraction of all the matter slowed the expansion. The end result is that both $\rho$ and $\dot a$ started high and decreased with time.
So it is not the case that a high matter density causes a high value of $\dot a$, but rather that in an expanding universe they cannot help but be correlated. The first Friedmann equation tells us how they are correlated.
A: You ignored the $k$ term, but it's crucial here. $k$ is the curvature not of spacetime but of constant-$t$ spatial slices, so it depends not only on spacetime curvature (represented by $ρ$ and $Λ$) but also on the extrinsic curvature of the spatial slice in the spacetime (represented by $\dot a/a$). You can think of this equation as showing the relationship between the spatial curvature that appears in the metric and the physical spacetime curvature.
In the special case $Λ=ρ=0$, the equation becomes $\displaystyle H^2 = \frac{-kc^2}{a^2}$, which implies $R = c/|H|$, where $R=\sqrt{|k|}/a$ is the radius of curvature. ($k/a^2$ is the Gaussian curvature.)
If also $p=0$, then there is no Riemann curvature, the spacetime is Minkowski space, and this equation is easy to interpret: the radius of curvature equals the time since the big bang (or until the big crunch). This makes sense because surfaces of constant $t$ are surfaces of constant distance from the $t\to0$ limit of all comoving worldlines, which is a single point in Minkowski space. In Euclidean space, the points at a distance $R$ from a point form a sphere with curvature radius $R$; in Minkowski space the points at a timelike distance $R/c$ form a hyperbolic plane with curvature radius $R$.
The spatial curvature has, in this special case, no physical significance: you can cover the same region of spacetime with many different FLRW charts with different values of $k$ and $H$. It's a coordinate artifact.
When $ρ\ne 0$, the symmetry is broken by the presence of matter, and the FLRW coordinates are dictated by the broken symmetry, so it is not a pure coordinate artifact, but the equation still expresses the relationship between inherent curvature and the curvature of a particular coordinate system, and not a physical force.
