How does an oscillating charge in an dipole antenna cause the emission of an electric field?

I understand electric fields and magnetic fields are "two sides of the same coin." But I'm not sure how the alternation of positive and negative charges on a dipole antenna creates an emission of a electric field which will be used as a signal. Are the electrons in the metal atoms of the antenna being jostled in a group to make them "spit" out radiation? Or is the magnetic field expanding and collapsing "spitting" out the radiation similar to how inductors store and release energy? For context, here is a video that explains oscillating charges in dipole antenna: https://www.youtube.com/watch?v=-1TwrwNF0C8

Your question is at the core of 'antenna theory' and is a result of Maxwell's equations. You can think of such radiation in terms of E or H, since they alternate together. The description below might help you imagine how radiation happens.

You can think of the magnetic field lines as closed loops always (since their divergence is always zero: $\text{Div}(H)=0$, as we haven't known magnetic charges to exist separably). The electric field lines, however, may be open or closed. Open scenarios: may start from +ve charge and land on -ve charge (if we have such charges available near us); may start at infinity and land on -ve charge (if only -ve charges are available near us); may start from +ve charge and land at infinity (if only +ve charges are available near us). Closed scenario: when no charges (sources) are around, the divergence is zero, $\text{Div}(E)=0$, just like in the magnetic field.

Once you understand this picture, you can roughly imagine that the E field lines near a source like a dipole antenna (whose current is oscillating in time) are initially of the open loop type, and once they travel off to free space far enough (in the so-called 'far-field' of the antenna) from the dipole, they start to close up on themselves and alternate in a stand-off with the H loops, one feeding the other (like a dance), while travelling in space.

Equivalently, this can be seen to happen step-by-step by observing the interplay between the other two (curl) equations of Maxwell's, starting from the E field (with charges as sources) or the H field (with current as source). If we start from H we see that: the time variation of charges in the antenna's wire is the current ($J$) which gives rise to H field curl (i.e. closed loops): $\text{Curl}(H)=J$ (if we assume $E=0$ at the source). The varying H field around the wire then, in turn, excites the E field curl, which also forms closed loops around the H loops: $\text{Curl}(E)=-\mu\partial H/\partial t$. Thus, you can say that the current in the dipole initiated the H field into the air in its vicinity, which then took things forward by exciting the E field, which then excite new H loops, then E loops, and so on, to propagate into space. (If you describe the same picture after quarter period in time, which is half the period of the standing wave on the wire, the same thing will happend but now switch E and H). If we start from E we see that: the current in the wire means +ve and -ve charges at the dipole terminals (like an open capacitor), which makes the E field stretch from one arm to another at a given moment, and such E field varies in strength with the current cycle, as the charges swing. For a complete cycle, the electric field forms a complete loop and, in turn, gives rise to an H loop, and so on... The two descriptions are equivalent and when H is max E is min, and vice versa.

For a Hertizian dipole (length=half wavelength), the first loops form and 'break off' from the antenna (like the soap bubbles that children blow into a soap ring) roughly at the distance of half wavelength from the dipole antenna, and roughly within a time duration of half period (i.e. wavelength travel time in space). It might be tricky to imagine how the E field manages to 'let go' of the antenna's metal, but you can try to imagine how the charge builds up positively at the dipole arms in time, and then decreases in amount to swing to the negative charge in each cycle. As the cycle crosses from +ve charge towards -ve charge (and vice versa), its passes through zero charge, which momentarily allows the close field lines to 'detach' from the chargeless wire to form its closed loop (or 'bubble'). The next cycle generates the next 'bubble', and so on.

Graphical explanation certainly helps the imagination here. A good way to see this clearly is to look at pages 9-17 in Balanis' great classic textbook (these pages can be seen actually from the review pages available on Amazon, even if you don't buy the book).

As the current keeps varying, such waves keep forming and radiating out of the antenna. They then reach the far-field zone (roughly away from the antenna by a radius of the order of the wavelength) which basically means they have become independent from their radiating antenna structure (i.e. non-reactive field) and can convey real average power to a load (another antenna that later face them). Therefore, they have linked two separate antennas, and can be called a 'signal'. Since their frequency content (bandwidth) can be modulated to carry information, such signal can carry information and is the basic element of wireless communications. Note that the bandwidth is what carries information, not the centre frequency (carrier frequency), which is simply a 'dull' sinusoial wave. The centre frequency, however, is usually chosen to be a high radio frequency (RF), such as 1-3 GHz in modern mobile devices, because it brings down the wavelength to a few centimeters and, since the dipole is half that in size, results in antennas of convenient size. Other advantages include having more channels (bandwidths$<<$centre freuqncy) for multiples users as the frequency is higher, but that is not the physical reason why it must be high frequency (which is really to keep the antenna size reasonably manageable).