Can we assume the second photo is showing us the ends of the existing speaker wires, and not the wires you intend to splice?

The local hardware or building supply store will have several methods of splicing wire.

The most common and easy would be the Insulated Butt Splice -

https://www.homedepot.com/p/Gardner-Bender-22-10-AWG-Butt-Splice-Heat-Shrink-Assortment-24-Pack-GBX-TK/205861923

The Color of the Butt Splice is related to the size of wire it will accommodate.

They are crimped with a common tool like this -

https://www.homedepot.com/p/Gardner-Bender-Terminal-and-Crimping-Tool-Kit-GS-67K/202518592

These do come in kits with both Crimp Connectors and a Crimp Tool. The Crimp Tool listed above is part of a kit.

https://www.amazon.com/s?k=crimp+tool+kit

Here is a UK link to more options -

Splicing Speaker Wires - Quick and Easy

https://www.avforums.com/threads/splicing-speaker-wires-quick-and-easy.2234425/

Everything linked to in that Post is available in the USA.

Example, - Banana Female to Female Coupler -

https://www.amazon.com/TKDMR-Insulated-Connectors-Extension-Adapters/dp/B07X7LJRNX/