What does it mean when a ticket is under "airport control"?

So my main question is, what is “airport control” and why does it happen?

The definition may be found in IATA Resolution 722f.

Airport Control — Indicates the Marketing or Operating Carrier has secured the electronic ticket flight coupon(s) prior to the scheduled departure. This control feature enables the Carrier holding airport control to update the coupon status indicator at the carrier/local airport level and not be dependent upon sending/receiving a link authorisation to process the passenger.

An air ticket can be honoured by a different airline who issued it. That is the key reason for standardisation. It is also why the notion of control is important. With a paper ticket, the operating carrier simply detached the relevant flight coupons from the ticket at check in time and took the passenger coupon from you when you boarded the plane. That was their proof to the issuing carrier that they had rendered transportation under the ticket. Your possession of a flight coupon was proof that you had paid for transportation and not used it yet. The electronic equivalent to physical possession of paper flight coupons is the idea of “control”.

Control establishes possession of the Electronic Ticket flight coupons. […]
The Validating Carrier shall transfer control of an ET flight coupon to only one carrier at any given time. Transfer of control of a flight coupon by the Validating Carrier shall constitute endorsement of that coupon under the provisions of Resolution 736.

Once the coupon falls under airport control, the operating carrier may check the passenger into the flight (changing the coupon status to C), redeem the value of the coupon (changing the coupon status to L, “lifted”) when the passenger scans his boarding pass to board the plane, or refund the coupon back to the original form of payment. If you changed your mind about travelling on a fully refundable fare, it can change the coupon status back to O (“open for use”).

Later, the operating carrier returns control of the coupon, together with the appropriate status update, back to the issuing carrier.

Actually, I booked mine through a government travel agency and he did not

Not unusual for different booking channels to cause tickets to be issued in different ways. The operating carrier can always take ownership of a valid flight coupon at check in time, but it may be inhibited from doing so in advance. In your case (speculating) you may have fallen down a gap where the departure airport did not take control of the agency-issued coupon but the issuing agent was happy to allow the operating carrier to revalidate the coupon into a higher booking class.

While trimming the flight from space control (Revenue Management Team), they block all access while issuing the manifesto the that particular flight.
So, that time travel agent or traveller cant do any check in due to flight goes on airport control.

Airport Control generally means that now whatever amendment/changes/editing will be done by check in staff at Airport.

There is a lot of shuffling at the last minute before a flight is dispatched, because a lot of problems materialize on the scene. A couple with an infant from an earlier, canceled flight might be put on this one, but now require a bassinet, which is only available in certain seats. Someone broke his leg after booking the flight and now won’t fit in the original seat. There might be changes in cargo, or a last-minute purchase by a frequent flyer that has cascading effects on upgrade or standby precedence.

You can’t have the operations team at the airport setting one seat assignment only for the reservations center to reverse it a minute later. As such, every airline sets a cutoff time when a flight goes under airport control, which is to say that the authority to make certain changes is cut off from reservations (including phone agents, customer-facing websites and apps, and anything partner airlines have access to), and those changes can only be done by the operations team at the airport.

Sometimes a ticket is flagged as under airport control when all it really is is “read-only” to you. A common example is trying to modify flight details like seat or meal preferences via a third party, like a partner airline’s website, or a corporate travel portal, which can sometimes show requests but not make them. You can sometimes contact the operating airline directly to make those changes for you.

At other times, even the reservations team at the operating carrier can’t. For example, some airlines only allow you to request a seating preference (aisle/window, and in the olden days, smoking/non-smoking), not a seating assignment (i.e. for a specific seat), which is not made until the day of the flight itself. I encountered this some years back on Lufthansa, booked through pre-merger United, when a flight was listed as under airport control even though it was weeks before departure.