What do we mean by end-to-end communication?
Communication between two computers in the internet.
Note that the information stored in the TCP part of an IP packet is ignored by the routers between the two computers (at least if NAT is not used).
What is the role of that connection?
To understand this, you'll first have to think about how the internet works:
IP packets are sent from one computer to another computer. The IP protocol itself does not have any possibility for the sending computer to find out if the packet really reached the destination computer.
IP packets can be routed in a way that the second packet arrives at the destination computer before the first one if two packets are sent.
The length of an IP packet is limited; in the worst case the limit is about 500 bytes (payload without headers) for IPv4 and about 1200 bytes for IPv6. In the "real-world" a limit of about 1500 bytes is often found.
Now think about transmitting some text (similar to an E-Mail) of 3000 bytes size between two computers that allow an IP packet size of 1500 bytes. You'll have to send that text in two IP packets. The receiving computer will not know the order of the two parts of the text and it will not know if the text received is complete because a third packet may have been lost.
The sending computer will not know if the receiving computer has received the two parts of the text in the correct order and it will also not know if one of the two parts has been lost.
If you send two texts, the receiver won't know if you sent one large text of 6000 bytes size or two texts of 3000 bytes size.
A transport protocol (such as TCP) will add some extra information to the IP packet that contains all the necessary information for the receiver. Many transport protocols (this is also true for TCP) will send packets back to the sender so that the sender is informed about the fact that the message has been received correctly.
A "connection" provided by some transport protocol allows you to exchange data between two devices in some network the same way data is exchanged between two telephones during a telephone connection:
If a large amount of data (or voice) is "sent" by one telephone, it will be "received" by the other telephone.
Couldn't reliability be assured without the connection?
This would mean that you would have to use another transport protocol than TCP. Such protocols exist (especially in non-IP networks; example: TPX).
However with a very high probability such a protocol will also work in a way that you can call this a "connection".
So you would only replace TCP by another protocol that works similarly.
That connection is not physical, ... Is that concept correct?
Right. The word "connection" in the context of transport protocols has a totally different meaning than the word "connection" in the context of OSI layer-1.
"All packets will be sent from the same connection" ... what do is it trying to say?
Without reading the rest of the text this is difficult to say.
But think about the following situation:
You open your web browser twice and in both windows you open the URL "stackexchange.com". Because your provider's network is currently overloaded, loading the web site takes a long time.
In this case each web browser window will establish a TCP connection to the server "stackexchange.com". This connection will be active until loading the web site has been finished.
In other words: There are two TCP connections between your computer and the server the same time!