High pressures and low pressures at any height do not exist in isolation. Had to remind myself a lot of this (so anyone feel free to correct), but the first thing worth highlighting is that the Bermuda High is a warm core high pressure and so extends a great depth through the atmosphere.
This image is a fairly typical cross section of the atmosphere at 35 North in June:
Source: NCEP/NCAR Reanalysis Monthly Means
The left of the two dashed lines is the center of the Bermuda High\Ridge. Note how it it is very upright and strong through a large height... so the winds at low levels tend to be aligned to the winds at higher levels. The high at each level has the same cause... being a favored area for the consistent sinking air caused by the Hadley Circulation [i.e. the necessity of poleward moving air to return to the Equator once it reaches certain velocities]
On the other hand the blue trough\low (the dashed line further to the right) is tilted. This is the typical structure of midlatitude cyclones (i.e. cold core lows). The starting upper level feature causes the surface low to develop ahead of it (downstream) because the air moving out of the upper trough sparks rising motion below it due to divergence\positive vorticity advection increasing with height.
The result (in the NH) is there will tend to be northwesterly winds at the ground locations near where the axis of the trough is passing over (and so southwest of the surface low\northeast of the surface high). Such northwesterly winds will tend to bring cooler air and lower relative humidities in most places, but it just how cold or dry the air will be depends upon the source region of the airmass the northwest winds are blowing in (which itself relates in large part back to the strength and larger history of the trough itself).
Overall, fundamental cause of connections between motion\pressures at different atmospheric heights is the fact mass continuity must occur (if air rises, it must be replaced, and it must eventually sink somewhere else to maintain the weight-density to prevent continuing mass buildup/loss), and works out to clearer relationships such as thermal wind (the reality that different winds at different height levels is a direct consequence\identity of temperature advection in the layer between the levels) and the fact the two main terms that cause vertical motion in the quasigeostrophic approximation are differential vorticity advection with height (i.e. different spin changes at different levels causes vertical movement) and maxima of temperature advection below (plus two other terms on friction and external heat).
In a sense, the fact pressure at one elevation induces changes\motion in another elevation maybe shouldn't seem any less weird than the fact that a low-level low pressure system can affect the wind and weather hundreds of miles away from it horizontally. This isn't spooky action at a distance, this is a continuous fluid where changes to one part of it causes impacts on another part.