You obviously are using a CC version and your subscription is not valid anymore.
If you don't have a subscription you need to either install the perpetual version (LR5) or turn your test version (LR CC 2015) into a stand-alone version (LR6): https://helpx.adobe.com/lightroom/kb/serialize-lightroom-cc-trial-to-activate-as-lightroom -6.html
If you havea valid subscription log out and back in: Sign in, Sign out | Creative Cloud desktop app
For more precise instructions, you need to tell us which version you have installed and for which version you own a license or a subscription.
If you bought a serial-number-licensed LR a year ago it would be LR 5.
To install that on a second computer (your new mac) you should uninstall whatever LR version you have on their now, then download and install the LR 5.7.1 update from the Adobe Updates page: