Structural glasses feature quasilocalized excitations whose frequencies ω follow a universal density of states D(ω)∼ω4. Yet, the underlying physics behind this universality is not yet fully understood. Here we study a mean-field model of quasilocalized excitations in glasses, viewed as groups of particles embedded inside an elastic medium and described collectively as anharmonic oscillators. The oscillators, whose harmonic stiffness is taken from a rather featureless probability distribution (of upper cutoff κ0) in the absence of interactions, interact among themselves through random couplings (characterized by strength J) and with the surrounding elastic medium (an interaction characterized by a constant force h). We first show that the model gives rise to a gapless density of states D(ω)=Agω4 for a broad range of model parameters, expressed in terms of the strength of stabilizing anharmonicity, which plays a decisive role in the model. Then --- using scaling theory and numerical simulations --- we provide a complete understanding of the non-universal prefactor Ag(h,J,κ0), of the oscillators' interaction-induced mean square displacement and of an emerging characteristic frequency, all in terms of properly identified dimensionless quantities. In particular, we show that Ag(h,J,κ0) is a non-monotonic function of J for a fixed h, varying predominantly exponentially with −(κ0h2/3/J2) in the weak interactions (small J) regime --- reminiscent of recent observations in computer glasses --- and predominantly decays as a power-law for larger J, in a regime where h plays no role. We discuss the physical interpretation of the model and its possible relations to available observations in structural glasses, along with delineating some future research directions.