Sulphates of alkali earth metals dissolve like this:
$$ce{MSO4(s) <=>[H2O] M^2+(aq) + SO4^2-(aq)}tag{1}$$
with the solubility product constant (kept original eq. numbering):
$$K_mathrm{sp} = a(ce{M^2+})a(ce{SO4^2-}),tag{3}$$
where $a(ce{X})$ is thermodynamic activity of the ion $ce{X}$. For diluted solutions is can be approximated by molar concentrations (denoted as []) as
$$K_mathrm{sp} = [ce{M^2+}][ce{SO4^2-}]tag{4}.$$
- $K_mathrm{sp} gt [ce{M^2+}][ce{SO4^2-}] implies text{net dissolution}$
- $K_mathrm{sp} lt [ce{M^2+}][ce{SO4^2-}] implies text{net precipitation}$
As a hydrogensulphate anion is not strong acid ($mathrm{p}K_mathrm{a}=1.99$) , concentrated hydrochloric acid protonizes in large extent sulphate to hydrogensulphate.
$$ce{SO4^2-(aq) + H+(aq)<=>>[conc. HCl] HSO4-(aq)}tag{2}$$
Decreasing of sulphate concentration disbalances dissolution equilibrium and there is ongoing net dissolution until the equilibrium is reached again.
$ce{CaSO4}$ is slightly soluble ($pu{0.26 g/100 ml}$ at $pu{25 ^{circ}C}$ (dihydrate)), therefore well soluble in $ce{HCl}$.
Solubility of $ce{BaSO4}$ is much less, ($pu{0.2448 mg/100 mL}$ at $pu{20 ^{circ}C}$ ) so it needs (hot) concentrated $ce{H2SO4}$. Concentrated $ce{H2SO4}$ acts here as the solvent. $pu{2-4%}$ of $ce{H2O}$ is practically all converted to $ce{H3O+(solv)}$. Additionally, $ce{H2SO4}$ partially autodissociates.
begin{align} ce{H2SO4(l) + H2O &-> HSO4-(solv) + H3O+(solv)}tag{5} ce{2 H2SO4(l) &<=> HSO4-(solv) + H3SO4+(solv)}tag{6} end{align}
This causes extremely high activity of solvated $ce{H+}$, extremely low activity of $ce{SO4^2-(solv)}$,
begin{align} ce{SO4^2-(solv) + H3O+(solv) &-> HSO4-(solv) + H2O(solv)}tag{7} ce{SO4^2-(solv) + H2SO4(l) &-> 2 HSO4-(solv)}tag{8} ce{SO4^2-(solv) + H3SO4+(solv) &-> HSO4-(solv) + H2SO4(l)}tag{9} end{align}
leading to relatively high $ce{BaSO4(s)}$ solubility.
It can be also described as
$$ce{BaSO4(s) + H2SO4(l) <=>[H2SO4] Ba(HSO4)2(solv)},$$
in a way analogical to:
$$ce{CaCO3(s) + H2O(l) + CO2(aq)) <=> Ca(HCO3)2(aq)}$$