"Book Separately" means those tickets are sold separately, and you have to book tickets for each of segment of the travel. Such flights are indicated with a separate green line.
In this situasion, you might have to claim luggages, clear the customs and immigration(for international flights) upon arrival, and re-checkin for next flight. For each segment of the entire flight, you might need multiple visa or local immigration documents before entering the country. Be aware, If your last flight was delayed too long that check-in counter or gate for next flight was closed, you might be considered as "no-show."
We suggest that you check on Airline's website, or local authority like DOT for your rights and regulations, and consider transfer time between flights before booking tickets.
Generally speaking, Airline companies sell all segments of flights in the same ticket, which means you will receive only one flight booking number. Such tickets are displayed with a connected green line.
However, only few airlines provide direct transfer services in some specific stops (e.g., Scoot's Scoot-Thru, and AirAsia's FLY-THRU) for eliminating usual worries around connecting between flights. Please note that some airlines will not guarantee your transfer whether the flights are issued on one ticket or not.