$k_2$ is one of three tidal Love-Shida numbers related to how gravitation of another body (Jupiter in this case) changes a planet-like body's second degree spherical harmonics (Io in this case).
Three Love and Shida numbers exist for each degree of spherical harmonic coefficients. The three Love-Shida numbers for a given degree $n$ are
- $k_n$, which describes how the other body changes the $n^{th}$ degree spherical harmonic coefficients of the body of interest. This makes the spherical harmonic coefficients time-dependent.
- $h_n$, which describes how those changes in the $n^{th}$ degree spherical harmonic coefficients affect the height of the body of interest relative to the tide-free shape.
- $l_n$, which describes how those changes in the $n^{th}$ degree spherical harmonic coefficients result in horizontal shifts of the body of interest relative to the tide-free shape.
A.E.H. Love described the Love numbers $k_n$ and $h_n$ in a 1909 paper to aid in describing the Earth's solid body tides. T. Shida added $l_n$ to the mix a bit later (1912). The zeroth order Love numbers are in a sense baked-in to the spherical harmonics model. The second order Love and Shida numbers are the key numbers of interest as the time-varying changes to the second degree spherical harmonic coefficients strongly dominate over higher degree terms.
I'm not going to attempt to reproduce the math-out equations here. You can find the math-out equations in various papers at iers.org. (The concept was developed to help in explaining the Earth tides, so papers at iers.org make imminent sense.) All of the Love and Shida numbers are unitless. A body made of the perfectly rigid form of unobtanium will have a $k_2$ value of 0, while a body made of the incompressible dust form of unobtanium will have a $k_2$ value of 1.5. Real objects will have a $k_2$ value between 0 and 1.5.
How does it relate to Io's volcanism?
There are multiple models regarding the details of Io's volcanism. Each of these different models results in different plausible range of $k_2$. Estimated values for $k_2$ based on Juno gravity experiments can thus rule out some of these models. By way of analogy, scientists long thought that the Moon and Mars had completely solid cores. Gravity experiments ruled out those concepts. The Moon and Mars have molten or partially molten cores (solid inner cores and a molten outer cores). The observed $k_2$ values from gravity experiments are inconsistent with the solid core hypothesis for those two bodies but are consistent with the molten / partially molten core hypothesis.
How can Juno constrain its value?
Juno by itself cannot do so. Juno's telecommunication system is outfitted with equipment that enables the Deep Space Network to track Juno's range and range rate with extreme precision. (The range rate measurements are the ridiculously precise.) With multiple close fly-bys of Io, those precise measurements give the Juno gravity science team the data needed to estimate Io's second degree spherical harmonics coefficients, and hence Io's $k_2$ value.