What do they mean when they say, "the trouble is in the Central Office".
In telephone communication in the United States, a Central Office (CO) is an office in a locality to which subscriber home and business lines are connected on what is called a local loop. The Central Office has switching equipment that can switch calls locally or to long-distance carrier phone offices.

Would you explain to me what a dial up modem is?
Dial-up modems pertain to a telephone connection in a system of many lines shared by many users. A dial-up connection is established and maintained for a limited time duration. The alternative is a dedicated connection, which is continuously in place. Dial-up lines are sometimes called switched lines and dedicated lines are called non-switched lines. A dedicated line is often a leased line that is rented from a telephone company. A dial-up connection can be initiated manually or automatically by your computer's modem or other device.

What is the Local Loop?
In Telephony, a Local Loop is the wired connection from a Telephone Company's Central Office in a locality to its customers' telephones at homes and businesses. This connection is usually on a pair of copper wires called twisted pair. The system was originally designed for voice transmission only using analog transmission technology on a single voice channel. Today, your computer's modem makes the conversion between analog signals and digital signals. With Integrated Services Digital Network ISDN or Digital Subscriber Line (DSL), the local loop can carry digital signals directly and at a much higher bandwidth than they do for voice only.

What does Analog Mean?
In telecommunications, an analog signal is one in which a base carrier's alternating current frequency is modified in some way, such as by amplifying the strength of the signal or varying the frequency, in order to add information to the signal. Broadcast and telephone transmission have conventionally used analog technology. An analog signal can be represented as a series of sine waves. The term originated because the modulation of the carrier wave is analogous to the fluctuations of the human voice or other sound that is being transmitted. Analog describes any fluctuating, evolving, or continually changing process.